Hawkes process and Edgeworth expansion with application to maximum likelihood estimator

Abstract

We provide a rigorous mathematical foundation of the theory for the higher-order asymptotic behavior of the one-dimensional Hawkes process with an exponential kernel. As an important application, we give the second-order asymptotic distribution for the maximum likelihood estimator of the exponential Hawkes process.

Appendix

Hereafter, when we write as $X(T) \lesssim T^a$ for $a \in {\mathbb {R}}$, it means that there exist positive constants C and $T'$ such that $X(T) \le CT^a$ holds for any $T>T'$.

1.1 Proofs of Section 2.1

To prove Theorem 2.1, we should give the asymptotic expansion for the characteristic function of $S_T$. The following discussion is a rework of Götze and Hipp (1983) and Yoshida (2004) to a form allowed when the variance is non-degenerate.

First, we introduce some notations. Let $N(T) = \lfloor T \rfloor + 1$ and divide the interval [0, T] into intervals $\{I_i\}_{i= 0, \dots , N(T)}$ such that $I_0 = [0, 0]$, $I_i = [i-1,i]$ for $i= 1, \dots , N(T) -1$ and $I_{N(T)} = [N(T)-1, T]$. Denote $Z_{I_i}$ as $Z_i$ for any $i = 0, \dots , N(T)$.⁵ There exists a smooth function $\phi : {\mathbb {R}}^d \rightarrow [0,1]$ such that $\phi (x) = 1$ if $|x| \le 1/2$, and $\phi (x)=0$ if $|x| \ge 1$. Choose a positive constant $\beta \in (0, \frac{1}{2})$ and put $\phi _T(x) = x\phi \big (x/2T^{\beta }\big )$. Then $\phi _T(x) = x$ if $|x| \le T^{\beta }$ and $\phi _T(x) = 0$ if $|x| \ge 2T^{\beta }$. Let $Z^*_i = \phi _T(Z_{i}) - E[\phi _T(Z_{i})]$ for any $i = 0, \dots , N(T)$ and $S_T^* = T^{-\frac{1}{2}}\sum _{i = 0}^{N(T)} Z^*_i$. Write the characteristic function of $S^*_T$ by $H_T(u) = E[e^{iu'S_T^*}]$ for $u \in {\mathbb {R}}^d$. For random variables X and V, we define $E[X](V) = E[Xe^{iV}]/E[e^{iV}]$. Define the cumulant of real-valued random variables $X_1, \dots , X_r$ shifted by a random variable V as

$$\begin{aligned} \kappa \left[ X_1, \dots , X_r \right] (V) =\frac{\partial ^r}{\partial \varepsilon _1 \cdots \partial \varepsilon _r}\bigg |_{\varepsilon _1 = \cdots = \varepsilon _r = 0} \log \bigg ( E\big [\exp \big (i\varepsilon _1X_1 + \cdots + i\varepsilon _rX_r \big )\big ](V)\bigg ) \end{aligned}$$

and write $\kappa \left[ X_1, \dots , X_r \right] = \kappa \left[ X_1, \dots , X_r \right] (0)$. In this subsection, we assume that the conditions [A1] and [A2] hold. By using the mixing property, it is possible to evaluate cumulants as follows. Write the i-th element of a vector X as $X^{(i)}$.

Proposition 6.1

Let $\bar{L} > 0$. Set $r \in {\mathbb {N}}$ with $r \le \bar{L}$ and $a_1, \dots , a_r \in \{1, \dots , d\}$. Then, for any $\varepsilon > 0$, there exists $\delta \in (0,1)$ such that

$$\begin{aligned}&1_{\left\{ |u| < T^{\delta } \right\} }(u) \left| \kappa \left[ S_T^{* (a_1)}, \dots , S_T^{* (a_r)} \right] \left( \eta u' S_T^*\right) \right| \\&\lesssim T^{-\frac{r-2}{2} + \varepsilon (r-1)} \quad \text {uniformly in}\ u \in {\mathbb {R}}^d\ \text {and}\ \eta \in [0,1]. \end{aligned}$$

Proof

It follows in a similar way as the proof of Lemma 5 in Yoshida (2004). $\square$

The next proposition is similar to Lemma 6 in Yoshida (2004). However, our assumption [A2] is stronger than the assumption in Yoshida (2004), thus we may take an arbitrary $L_3 > 0$ as the following.

Proposition 6.2

For any $L_3 > 0$, $r \in {\mathbb {N}}$ and $a_1, \dots , a_r \in \{1, \dots , d\}$,

$$\begin{aligned} \left| \kappa \left[ S_T^{* (a_1)}, \dots , S_T^{* (a_r)} \right] - \kappa \left[ S_T^{(a_1)}, \dots , S_T^{(a_r)} \right] \right| \lesssim T^{ - L_3\beta }. \end{aligned}$$

Proof

We immediately get

$$\begin{aligned}&\left| \kappa \left[ S_T^{* (a_1)}, \dots , S_T^{* (a_r)} \right] -\kappa \left[ S_T^{(a_1)}, \dots , S_T^{(a_r)} \right] \right| \\&\quad \le T^{-r/2}\sum _{0 \le j_1, \dots , j_r \le N(T)} \left| \kappa \left[ Z_{j_1}^{*(a_1)}, \dots , Z_{j_r}^{*(a_r)}\right] -\kappa \left[ Z_{j_1}^{(a_1)}, \dots , Z_{j_r}^{(a_r)}\right] \right| \\&\quad \le T^{-r/2}\sum _{0 \le j_1, \dots , j_r \le N(T)} \sum _{l = 1}^r \sum _{\begin{array}{c} \alpha _1, \dots , \alpha _l;\\ \alpha _1 + \dots + \alpha _l = \{1, \dots , r\} \end{array}} \frac{(-1)^{l-1}}{l} \bigg | \prod _{m = 1}^l E \left[ \prod _{i \in \alpha _m} \phi ^{(a_i)} _T(Z_{j_i})\right] \\&\qquad - \prod _{m = 1}^l E \left[ \prod _{i \in \alpha _m} Z_{j_i}^{(a_i)} \right] \bigg |. \end{aligned}$$

Moreover,

$$\begin{aligned}&\left| \prod _{m = 1}^l E \left[ \prod _{i \in \alpha _m} \phi ^{(a_i)} _T(Z_{j_i})\right] -\prod _{m = 1}^l E \left[ \prod _{i \in \alpha _m} Z_{j_i}^{(a_i)} \right] \right| \\&\quad \le \sum _{m = 1}^l \left( \prod _{m' = 1}^{m - 1} \left| E \left[ \prod _{i \in \alpha _{m'}} \phi ^{(a_i)} _T(Z_{j_i})\right] \right| \right) \left| E \left[ \prod _{i \in \alpha _m} \phi ^{(a_i)} _T(Z_{j_i}) -\prod _{i \in \alpha _m} Z_{j_i}^{(a_i)} \right] \right| \\&\qquad \left( \prod _{m' = m + 1}^{l} \left| E \left[ \prod _{i \in \alpha _{m'}} Z_{j_i}^{(a_i)} \right] \right| \right) , \end{aligned}$$

and the condition [A2] yields

$$\begin{aligned}&\left| E \left[ \prod _{i \in \alpha _m} \phi ^{(a_i)} _T(Z_{j_i}) -\prod _{i \in \alpha _m} Z_{j_i}^{(a_i)} \right] \right| \\&\quad = \left| \sum _{k'=1}^k E \left[ \left( \prod _{i =1}^{k'-1} \phi ^{(a_i)} _T(Z_{j_i})\right) \left( \prod _{i =k'+1}^{k} Z_{j_i}^{(a_i)} \right) \left( \phi ^{(a_{k'})} _T(Z_{j_{k'}}) - Z_{j_{k'}}^{(a_{k'})} \right) 1_{\left\{ \left| Z_{j_{k'}}^{(a_{k'})} \right| \ge T^{\beta } \right\} } \right] \right| \\&\quad \le \bigg | \sum _{k'=1}^k T^{ - L_3 \beta } E \left[ \left( \prod _{i =1}^{k'-1} \phi ^{(a_i)} _T(Z_{j_i})\right) \left( \prod _{i =k'+1}^{k} Z_{j_i}^{(a_i)} \right) \right. \\&\qquad \left. \left( \phi ^{(a_{k'})} _T(Z_{j_{k'}}) - Z_{j_{k'}}^{(a_{k'})} \right) \left| Z_{j_{k'}}^{(a_{k'})} \right| ^{L_3}1_{\left\{ \left| Z_{j_{k'}}^{(a_{k'})}\right| \ge T^{\beta } \right\} }\right] \bigg | \lesssim T^{ - L_3 \beta }. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \kappa \left[ S_T^{* (a_1)}, \dots , S_T^{* (a_r)} \right] -\kappa \left[ S_T^{(a_1)}, \dots , S_T^{(a_r)} \right] \right| \lesssim T^{ - L_3 \beta + \frac{r}{2}}. \end{aligned}$$

Since $L_3$ is arbitrary, we get the conclusion. $\square$

From Propositions 6.1 and 6.2, we immediately get the following statement.

Corollary 6.3

Let $\bar{L} > 0$. Set $r \in {\mathbb {N}}$ with $r \le \bar{L}$ and $a_1, \dots , a_r \in \{1, \dots , d\}$. Then, for any $\varepsilon > 0$,

$$\begin{aligned} \left| \kappa \left[ S_T^{(a_1)}, \dots , S_T^{(a_r)} \right] \right| \lesssim T^{-\frac{r-2}{2} + \varepsilon (r-1)}. \end{aligned}$$

We evaluate the gap between $H_T(u)$ and its expansion $\hat{\varPsi }_{T,p,D}(u)$. Allowing for the abuse of symbols, we define ${\mathbb {D}}$ as the derivative with respect to u in the same way as (2.9).

Proposition 6.4

Let $\bar{L}>0$. There exist $D>0$, $\delta > 0$ and $\delta _0 > d\delta$ such that

$$\begin{aligned} 1_{\left\{ |u| < T^{\delta } \right\} }(u) \left| {\mathbb {D}}^{\mathbb {n}} \left( H_T(u) - \hat{\varPsi }_{T,p,D}(u) \right) \right| \lesssim T^{-\frac{p-2}{2} - \delta _0} \end{aligned}$$

uniformly in $u \in {\mathbb {R}}^d$ and $\mathbb {n}\in \{1, \dots , d\}^l$ with $l \le \bar{L}$.

Proof

Denote $\kappa ^*[u^{\otimes r}](V) = \kappa \left[ u'S_T^*, \dots , u'S_T^* \right] (V)$, $\kappa [u^{\otimes r}](V) = \kappa \left[ u'S_T, \dots , u'S_T \right] (V)$, $\kappa ^*[u^{\otimes r}] =\kappa ^*[u^{\otimes r}](0)$ and $\kappa [u^{\otimes r}] =\kappa [u^{\otimes r}](0)$. We have

$$\begin{aligned} H_T(u) = \hat{\varPsi }^*_{T,p}(u) + R^*_{p+1}(u), \end{aligned}$$

where

$$\begin{aligned} \hat{\varPsi }^*_{T,p}(u)&= \exp \left( \chi _{T,2}(u) \right) \\&\qquad \left\{ 1 + \sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}} (-1)^j i^{r_1+ \dots + r_j} \frac{\kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_j+2}]}{j!(r_1+2)!\cdots (r_j+2)!} \right\} ,\\ R^*_{p+1}(u)&= \exp \left( \chi _{T,2}(u) \right) \\&\qquad \left\{ \sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \ge p-1\}} (-1)^j i^{r_1+ \dots + r_j } \frac{\kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_j+2}]}{j!(r_1+2)!\cdots (r_j+2)!} \right. \\&\qquad \left. + \sum _{j=1}^p \sum _{j'=0}^{j-1} \frac{1}{j'!} \Bigg (\begin{array}{l} j \\ j' \end{array} \Bigg ) \left( \sum _{r=3}^p \frac{i^r}{r!} \kappa ^*[u^{\otimes r}] \right) ^{j'} \left( R_{p+1}(u)\right) ^{j-j'} \right. \\&\qquad \left. + \left( \sum _{r=3}^p \frac{i^r}{r!} \kappa ^*[u^{\otimes r}] + R_{p+1}(u) \right) ^{p+1} \frac{1}{p!} \int _0^1(1-t)^p\right. \\&\qquad \left. \exp \left( t \sum _{r=3}^p \frac{i^r}{r!} \kappa ^*[u^{\otimes r}] + t R_{p+1}(u) \right) dt \right\} \end{aligned}$$

and

$$\begin{aligned} R_{p+1}(u) = \frac{i^{p+1}}{p!} \int _0^1 (1-s)^p \kappa ^*[u^{\otimes p+1}] (su'S_T^*) ds - \frac{1}{2}\left( \kappa ^*[u^{\otimes 2}] + \chi _{T,2}(u) \right) . \end{aligned}$$

First, we consider $\left| {\mathbb {D}}^{\mathbb {n}}\left( \hat{\varPsi }^*_{T,p}(u) - \hat{\varPsi }_{T,p,D}(u) \right) \right|$. From the definition, we have

$$\begin{aligned}&\left| {\mathbb {D}}^{\mathbb {n}}\left( \hat{\varPsi }^*_{T,p}(u) -\hat{\varPsi }_{T,p,D}(u) \right) \right| \\&\quad \lesssim \left| {\mathbb {D}}^{\mathbb {n}} \left[ \Big ( e^{ \chi _{T,2}(u)} -e^{ - \frac{1}{2} u' \varSigma _{T,D}u } \Big ) \right. \right. \\&\qquad \left. \left. \left\{ 1 + \sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}} \kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_j+2}] \right\} \right] \right| \\&\qquad + \ \Bigg |{\mathbb {D}}^{\mathbb {n}} \Bigg [ e^{ - \frac{1}{2} u' \varSigma _{T,D}u }\sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}} \\&\qquad \Big ( \kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_j+2}] - \kappa [u^{\otimes r_1+2}] \cdots \kappa [u^{\otimes r_j+2}] \Big ) \Bigg ] \Bigg |. \end{aligned}$$

In the following, we assume that u satisfies $|u| \le T^{\delta }$. We have

$$\begin{aligned} \Big | e^{ \chi _{T,2}(u) } - e^{ - \frac{1}{2} u' \varSigma _{T,D}u } \Big | \le \ \Big | e^{ \frac{1}{2}T^{-D}|u|^2 } - 1 \Big | \lesssim \ T^{-D + 2\delta }. \end{aligned}$$

(6.1)

For the first term, by applying Proposition 6.1, Corollary 6.3 and (6.1), we get

$$\begin{aligned}&\left| {\mathbb {D}}^{\mathbb {n}} \left[ \Big ( e^{ \chi _{T,2}(u) } -e^{ - \frac{1}{2} u' \varSigma _{T,D}u } \Big ) \left\{ 1 +\sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}} \kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_j+2}] \right\} \right] \right| \\&\quad \lesssim \left( 1 + \big |{\textit{Var}}[S_T]\big | + T^{-D} \right) ^{\sharp \mathbb {n}} T^{-D + \delta (2 + \sharp \mathbb {n})} \\&\qquad \sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}} T^{ (r_1 + \dots + r_j)\left( -\frac{1}{2} + \varepsilon + \delta \right) + j(\varepsilon + 2\delta )} \\&\quad \lesssim T^{-D + \varepsilon (p + \bar{L}) + \delta (2p + 2 + \bar{L})}. \end{aligned}$$

By Proposition 6.2 and Corollary 6.3, the second term is estimated as below;

$$\begin{aligned}&\Bigg |{\mathbb {D}}^{\mathbb {n}} \Bigg [ e^{ - \frac{1}{2} u' \varSigma _{T,D}u } \sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}} \\&\qquad \Big ( \kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_j+2}] -\kappa [u^{\otimes r_1+2}] \cdots \kappa [u^{\otimes r_j+2}] \Big ) \Bigg ] \Bigg | \\&\quad \lesssim \Bigg |{\mathbb {D}}^{\mathbb {n}} \Bigg [ e^{ - \frac{1}{2} u' \varSigma _{T,D}u } \sum _{j = 1}^p \sum _{r_1, \dots , r_j = 1}^{p-2} 1_{\{r_1 + \dots + r_j \le p-2\}}\\&\qquad \times \sum _{k=1}^j \kappa ^*[u^{\otimes r_1+2}] \cdots \kappa ^*[u^{\otimes r_{k-1}+2}] \Big ( \kappa ^*[u^{\otimes r_k+2}] -\kappa [u^{\otimes r_k+2}] \Big ) \kappa [u^{\otimes r_{k+1}+2}] \cdots \kappa [u^{\otimes r_j+2}] \Bigg ] \Bigg |\\&\quad \lesssim \left( 1 + \big |{\textit{Var}}[S_T]\big | + T^{-D} \right) ^{\sharp \mathbb {n}} T^{\delta \sharp \mathbb {n}} T^{- L_3 \beta + \varepsilon p + \delta (3p-2)} \lesssim T^{-L_3\beta + \varepsilon (p + \bar{L}) + \delta (3p - 2 + \bar{L}) }. \end{aligned}$$

Therefore, for any $\varepsilon$, $\delta$ and $\delta _0$, we can choose sufficiently large D and $L_3$ such that

$$\begin{aligned} \left| {\mathbb {D}}^{\mathbb {n}}\left( \hat{\varPsi }^*_{T,p}(u) -\hat{\varPsi }_{T,p,D}(u) \right) \right| \lesssim T^{-\frac{p-2}{2} - \delta _0}. \end{aligned}$$

Finally, we have to show that $\left| {\mathbb {D}}^{\mathbb {n}} \left( H_T(u) - \hat{\varPsi }^*_{T,p}(u) \right) \right| =\left| {\mathbb {D}}^{\mathbb {n}} R_{p+1}^*(u) \right| \lesssim T^{-\frac{p-2}{2} - \delta _0}$. However, it follows by the same method as the proof of Lemma 7 in Yoshida (2004). In particular, we can choose $\delta _0$ and $\delta$ with $\delta _0 > d\delta$ in this proof. $\square$

Referring to Götze and Hipp (1978), we will prove Theorem 2.1 by using the smoothness of a function f. Let $S'_T =T^{-\frac{1}{2}}\sum _{i = 0}^{N(T)} \phi _T(Z_i)$ and $e_T =T^{-\frac{1}{2}}\sum _{i = 0}^{N(T)} E[\phi _T(Z_i)]$. Note that

$$\begin{aligned} \left| E[\phi _T(Z_i)] \right| = \left| E[\phi _T(Z_i) - Z_i]\right| \le E[ |Z_i| 1_{\{|Z_i| > T^{\beta }\}}] \le T^{-n\beta }E[|Z_i|^{n+1}] \end{aligned}$$

for any $i = 0, \dots , N(T)$ and $n \in {\mathbb {N}}$. Therefore, [A2] yields

$$\begin{aligned} |e_T| \lesssim T^{-L_4} \end{aligned}$$

(6.2)

for an arbitrarily large constant $L_4 > 0$.

Proof

(Proof of Theorem 2.1) Let $\varGamma = \lceil \frac{p-1}{2\delta } \rceil$ for some $\delta \in (0,1)$ and $f \in {\mathscr {E}}(\varGamma , L_1, L_2)$. Since

$$\begin{aligned}&\left| E\left[ f( S_T )\right] - \int _{{\mathbb {R}}^d}f(z) p_{T,p,D}(z)dz \right| \\&\quad \le \Big | E\left[ f(S_T)\right] - E\left[ f(S'_T)\right] \Big |\\&\qquad + \left| E\left[ f(S'_T)\right] - \int _{{\mathbb {R}}^d}f(z + e_T) p_{T,p,D}(z)dz \right| \\&\qquad + \left| \int _{{\mathbb {R}}^d}f(z + e_T) p_{T,p,D}(z)dz - \int _{{\mathbb {R}}^d}f(z) p_{T,p,D}(z)dz \right| =: \varDelta _1 + \varDelta _2 + \varDelta _3, \end{aligned}$$

we only have to estimate $\varDelta _1, \varDelta _2$ and $\varDelta _3$.

First, we consider $\varDelta _1$. Let $\eta > 0$. We can assume that $L_1$ is even by retaking $L_1$ and $L_2$ that satisfy $\sup _{|\alpha | \le \varGamma }|\partial ^\alpha f(x)| \le L_2(1+|x|)^{L_1}$ for every $x \in {\mathbb {R}}^d$. We set $A = \{ |S_T| \le T^{\eta }\}$ and $B = \{ |S'_T| \le T^{\eta }\}$. Similarly to Lemma 3.3 in Götze and Hipp (1983), we get

$$\begin{aligned} \varDelta _1\lesssim & {} T^{\eta L_1}P\big [S_T \ne S'_T\big ] + E\big [|S_T|^{L_1} 1_{A^c}\big ] + E\big [|S'_T|^{L_1} 1_{B^c}\big ] \\\lesssim & {} T^{\eta L_1}P\big [S_T \ne S'_T\big ] + E\big [|S'_T|^{L_1} 1_{B^c}\big ] + \Big | E\big [|S_T|^{L_1}\big ] - E\big [|S'_T|^{L_1}]\big ]\Big |. \end{aligned}$$

Let $L > 0$ be an arbitrary large constant. Since $P\big [S_T \ne S'_T\big ] \le \sum _{i=0}^{N(T)}P\big [|Z_i| > T^{\beta }] \lesssim T^{-n\beta +\frac{1}{2}}$ for any $n \in {\mathbb {N}}$, $T^{\eta L_1}P\big [S_T \ne S'_T\big ] \lesssim T^{-L}$ holds. Since $|S'_T| \le |S^*_T| + |e_T|$, we have

$$\begin{aligned} E\big [|S'_T|^{L_1} 1_{B^c}\big ] \lesssim E \left[ |S^*_T|^{L_1} 1_{ \{ |S^*_T| > \frac{ T^{\eta } }{2} \} } \right] + |e_T|^{L_1}. \end{aligned}$$

(6.3)

The moment has the representation by cumulants; for any even $n \in {\mathbb {N}}$

$$\begin{aligned} E \left[ |S^*_T|^n \right] =\sum _{|\alpha | = n} \sum _{k = 1}^n \sum _{\begin{array}{c} \alpha _1, \dots , \alpha _k;\\ \alpha _1 + \dots + \alpha _k = \alpha \end{array}} \frac{\alpha !}{k!\alpha _1! \cdots \alpha _k!} \prod _{m = 1}^k \kappa _{\alpha _m}\left[ S^*_T\right] , \end{aligned}$$

where $\kappa _{\alpha _m}\left[ S^*_T\right] = (-i)^{|\alpha _m|}\partial ^{\alpha _m}\log E[e^{iu'S^*_T}]|_{u=0}$ and $\alpha ! = \alpha ^{1;}! \cdots \alpha ^{d;}!$ for $\alpha = (\alpha ^{1;}, \dots , \alpha ^{d;}) \in {\mathbb {Z}}_+^d$. From Proposition 6.1 and the representation of moments by cumulants, $E \left[ |S^*_T|^n \right] \lesssim T^{\varepsilon n}$ for any even $n \in {\mathbb {N}}$ and $\varepsilon > 0$. Therefore,

$$\begin{aligned} E \left[ |S^*_T|^{L_1} 1_{ \left\{ |S^*_T| > \frac{ T^{\eta } }{2} \right\} } \right] \le \left( \frac{T^{\eta }}{2}\right) ^{-n} E \left[ |S^*_T|^{L_1+n} \right] \lesssim T^{-\eta n + \varepsilon (L_1 + n)}. \end{aligned}$$

Equations (6.2), (6.3) and the above inequality lead $E\big [|S'_T|^{L_1} 1_{B^c}\big ] \lesssim T^{-L}$ by choosing $\eta > \varepsilon$ and sufficiently large n. Since we took $L_1$ as an even number, the representation of moments by cumulants leads

$$\begin{aligned}&\Big | E\big [|S_T|^{L_1}\big ] - E\big [|S'_T|^{L_1}]\big ]\Big |\\&\le \sum _{|\alpha | = L_1} \sum _{k = 1}^{L_1} \sum _{\begin{array}{c} \alpha _1, \dots , \alpha _k;\\ \alpha _1 + \dots + \alpha _k = \alpha \end{array}} \frac{\alpha !}{k!\alpha _1! \cdots \alpha _k!} \left| \prod _{m = 1}^k \kappa _{\alpha _m}\left[ S_T\right] -\prod _{m = 1}^k \kappa _{\alpha _m}\left[ S'_T\right] \right| . \end{aligned}$$

From the definition of the cumulant, $\kappa _{\alpha _m}\left[ S'_T\right] = \kappa _{\alpha _m}\left[ S^*_T\right]$ for $|\alpha _m| \ge 2$ and $\kappa _{\alpha _m}\left[ S'_T\right] = e_T$ for $|\alpha _m| = 1$. Thus, Propositions 6.1, 6.2, Corollary 6.3 and (6.2) yield

$$\begin{aligned}&\left| \prod _{m = 1}^k \kappa _{\alpha _m}\left[ S_T\right] -\prod _{m = 1}^k \kappa _{\alpha _m}\left[ S'_T\right] \right| \\&\quad \lesssim \left| \prod _{m = 1}^k \kappa _{\alpha _m}\left[ S_T\right] -\prod _{m = 1}^k \kappa _{\alpha _m}\left[ S^*_T\right] \right| + T^{-L}\\&\quad =\left| \sum _{k'=1}^k \Bigg ( \prod _{m = 1}^{k'-1} \kappa _{\alpha _m} \left[ S_T\right] \Bigg ) \left( \kappa _{\alpha _{k'}}\left[ S_T\right] - \kappa _{\alpha _{k'}}\left[ S^*_T\right] \right) \Bigg ( \prod _{m = k'+1}^k \kappa _{\alpha _m}\left[ S^*_T\right] \Bigg ) \right| + T^{-L}\\&\quad \lesssim T^{-\frac{1}{2}(|\alpha _1| + \cdots + |\alpha _k| - 2k) +\varepsilon (|\alpha _1| + \cdots + |\alpha _k| - k) - L_3\beta } + T^{-L} \lesssim T^{-L}. \end{aligned}$$

Therefore, we get $\varDelta _1 \lesssim T^{-L}$. Since L is an arbitrary constant, we get $\varDelta _1 \lesssim T^{-\frac{p-2}{2}}$.

Second, we estimate $\varDelta _2$. Write $h_{e_T}(z) = h(z + e_T)$ for any function h on ${\mathbb {R}}^d$. Denote the distribution of $S^*_T$ as $dQ^*_T$. Then, we can rewrite

$$\begin{aligned} \varDelta _2 =\left| \int _{{\mathbb {R}}^d}f_{e_T}(z) d\big (Q^*_T - \varPsi _{T,p,D} \big )(z)\right| \end{aligned}$$

For $z, u \in {\mathbb {R}}^d$, the Taylor’s theorem yields

$$\begin{aligned} f_{e_T}(z) = \sum _{\alpha ; |\alpha |\le \varGamma } \frac{\partial ^{\alpha }f_{e_T}(z + u)}{\alpha !}(-u)^{\alpha } + g_T^{-1}(z, u) \end{aligned}$$

where

$$\begin{aligned} g_T^{-1}(z, u) = \sum _{\alpha ; |\alpha | = \varGamma } (-u)^{\alpha } \frac{\varGamma }{\alpha !} \int _0^1 \nu ^{\varGamma } \Big ( \partial ^{\alpha } f_{e_T}(z + \nu u) - \partial ^{\alpha }f_{e_T}(z + u) \Big )d\nu . \end{aligned}$$

Let ${\mathcal {K}}$ be a probability measure on ${\mathbb {R}}^d$ such that $\int _{{\mathbb {R}}^d}|z|^{\bar{L}}d{\mathcal {K}}(z) < \infty$ for sufficiently large $\bar{L} > 0$ and its Fourier transformation $\hat{{\mathcal {K}}}(u)$ satisfies $\hat{{\mathcal {K}}}(u)=0$ if $|u| > 1$. (Such ${\mathcal {K}}$ exists. See Theorem 10.1 in Bhattacharya and Rao 1976.) Moreover, let $d{\mathcal {K}}_T(u) =d{\mathcal {K}}(T^{-\delta }u)$ and $d{\mathcal {K}}_{T, \alpha }(u) =u^{\alpha }d{\mathcal {K}}_T(u)$. We have

$$\begin{aligned}&\int _{{\mathbb {R}}^d}f_{e_T}(z) d\big (Q^*_T - \varPsi _{T,p,D} \big )(z) \nonumber \\&\quad = \sum _{\alpha ; |\alpha | \le \varGamma } \int _{{\mathbb {R}}^d \times {\mathbb {R}}^d} \frac{\partial ^{\alpha }f_{e_T}(z + u)}{\alpha !}(-u)^{\alpha } d \big (Q^*_T - \varPsi _{T,p,D} \big )(z) d{\mathcal {K}}_T(u)\nonumber \\&\qquad + \int _{{\mathbb {R}}^d \times {\mathbb {R}}^d} g_T^{-1}(z, u) d \big (Q^*_T - \varPsi _{T,p,D} \big )(z) d{\mathcal {K}}_T(u) \nonumber \\&\quad = \sum _{\alpha ; |\alpha | \le \varGamma } \frac{(-1)^{\alpha }}{\alpha !} \int _{{\mathbb {R}}^d} \partial ^{\alpha }f_{e_T}(x) d\left( {\mathcal {K}}_{T,\alpha }* \big (Q^*_T - \varPsi _{T,p,D} \big )\right) (x)\nonumber \\&\qquad + \int _{{\mathbb {R}}^d \times {\mathbb {R}}^d} g_T^{-1}(z, u) d\big (Q^*_T -\varPsi _{T,p,D} \big )(z) d{\mathcal {K}}_T(u). \end{aligned}$$

(6.4)

We know that $\partial ^{\alpha }f_{e_T}(x) = \partial ^{\alpha }f(x +e_T)$ and $|e_T|$ is bounded in T. Then, Lemma 11.6 Bhattacharya and Rao (1976) and well-known properties of Fourier transform lead

$$\begin{aligned}&\left| \sum _{\alpha ; |\alpha | \le \varGamma } \frac{(-1)^{\alpha }}{\alpha !} \int _{{\mathbb {R}}^d} \partial ^{\alpha }f_{e_T}(x) d\left( {\mathcal {K}}_{T,\alpha }*\big (Q^*_T - \varPsi _{T,p,D} \big )\right) (x) \right| \\&\quad \le \left( \sup _{\begin{array}{c} |\alpha | \le \varGamma \\ x \in {\mathbb {R}}^d \end{array}}\frac{\left| \partial ^\alpha f(x) \right| }{1 + |x|^{L_1}}\right) \left| \sum _{\alpha ; |\alpha | \le \varGamma } \frac{(-1)^{\alpha }}{\alpha !} \int _{{\mathbb {R}}^d} 1 + |x + e_T|^{L_1} d\left( {\mathcal {K}}_{T,\alpha }*\big (Q^*_T - \varPsi _{T,p,D} \big )\right) (x) \right| \\&\quad \lesssim \sum _{\alpha ; |\alpha | \le \varGamma } \max _{|\beta | \le L_1 + d + 1} \int _{{\mathbb {R}}^d} \left| \partial ^{\beta }\left( \hat{{\mathcal {K}}}_{T,\alpha }(u)\big (H_T(u) - \hat{\varPsi }_{T,p,D}(u) \big )\right) \right| du. \end{aligned}$$

Since $\hat{{\mathcal {K}}}_{T,\alpha }(u) = i^{-|\alpha |}\partial ^{\alpha }\hat{{\mathcal {K}}}_{T}(u)$, we have $\mathop {\mathrm {supp}}\nolimits \hat{{\mathcal {K}}}_{T,\alpha }(u) \subset \{ |u| < T^{\delta }\}$. Moreover, $\big |\partial ^{\beta } \hat{{\mathcal {K}}}_{T,\alpha }(u)\big | \le \int _{{\mathbb {R}}^d}|z|^{\alpha + \beta } d{\mathcal {K}}_T(z) \lesssim T^{-\delta (|\alpha | + |\beta |)}$ holds. Thus, from Proposition 6.4, we can choose $\delta \in (0,1)$ and $\delta _0 > d\delta$ such that

$$\begin{aligned}&\max _{|\beta | \le L_1 + d + 1} \int _{{\mathbb {R}}^d} \left| \partial ^{\beta }\left( \hat{{\mathcal {K}}}_{T,\alpha }(u)\big (H_T(u) - \hat{\varPsi }_{T,p,D}(u) \big )\right) \right| du\\&\quad \lesssim \max _{|\beta _1|, |\beta _2| \le L_1 + d + 1} \int _{\{|u| < T^{\delta }\}} \left| \partial ^{\beta _1}\hat{{\mathcal {K}}}_{T,\alpha }(u)\right| \left| \partial ^{\beta _2}\big (H_T(u) - \hat{\varPsi }_{T,p,D}(u) \big )\right| du\\&\quad \lesssim T^{-\frac{p-2}{2} - \delta _0 + d\delta } \lesssim T^{-\frac{p-2}{2}}. \end{aligned}$$

It means that

$$\begin{aligned} \left| \sum _{\alpha ; |\alpha | \le \varGamma } \frac{(-1)^{\alpha }}{\alpha !} \int _{{\mathbb {R}}^d} \partial ^{\alpha }f_{e_T}(x) d\left( {\mathcal {K}}_{T,\alpha }*\big (Q^*_T - \varPsi _{T,p,D} \big )\right) (x) \right| \lesssim T^{-\frac{p-2}{2}}. \end{aligned}$$

(6.5)

On the other hand, from Proposition 6.1, $|\varSigma _{T,D}| \le T^{\varepsilon }$ holds for any $\varepsilon >0$. Therefore, $\int _{{\mathbb {R}}^d } |z|^{L_1} p_{T,p,D}(z)dz \le T^{\frac{L_5 \varepsilon }{2}}$ for some constant $L_5 > 0$ which depends on p. Thus, by taking sufficiently small $\varepsilon$,

$$\begin{aligned}&\left| \int _{{\mathbb {R}}^d \times {\mathbb {R}}^d} g_T^{-1}(z, u) d\big (Q^*_T - \varPsi _{T,p,D} \big )(z) d{\mathcal {K}}_T(u) \right| \\&\quad = \bigg | \int _{{\mathbb {R}}^d \times {\mathbb {R}}^d} \sum _{\alpha ; |\alpha | = \varGamma } (-u)^{\alpha } \frac{\varGamma }{\alpha !} \int _0^1 \nu ^{\varGamma } \Big ( \partial ^{\alpha }f_{e_T}(z + \nu u) - \partial ^{\alpha }f_{e_T}(z + u) \Big ) \\&\qquad d\nu d\big (Q^*_T - \varPsi _{T,p,D} \big )(z) d{\mathcal {K}}_T(u) \bigg |\\&\quad \lesssim (T^{-\delta })^{\varGamma } \left( \sup _{\begin{array}{c} |\alpha | \le \varGamma \\ x \in {\mathbb {R}}^d \end{array}}\frac{\left| \partial ^\alpha f(x) \right| }{1 + |x|^{L_1}}\right) \bigg | \int _{{\mathbb {R}}^d \times {\mathbb {R}}^d} \int _0^1u^{\alpha } \nu ^{\varGamma } \Big ( 1 + |z + T^{-\delta }u + e_T|^{L_1} \Big )\\&\qquad d\nu d\big (Q^*_T - \varPsi _{T,p,D} \big )(z) d{\mathcal {K}}(u) \bigg |\\&\quad \lesssim T^{-\frac{p-1}{2}} \bigg | \int _{{\mathbb {R}}^d } 1 + |z|^{L_1} d\big (Q^*_T - \varPsi _{T,p,D} \big )(z) \bigg | \\&\quad \lesssim T^{-\frac{p-1}{2}} \left( E\left[ |S^*_T|^{L_1}\right] + \int _{{\mathbb {R}}^d } |z|^{L_1} p_{T,p,D}(z)dz \right) \lesssim T^{-\frac{p-2}{2}}. \end{aligned}$$

In conclusion, we get $\varDelta _2 \lesssim T^{-\frac{p-2}{2}}$ from (6.4), (6.5) and the above inequality.

Finally, we consider $\varDelta _3$. With the help of the mean value theorem, we can deduce as below; for some $\tau \in (0, 1)$,

$$\begin{aligned} \varDelta _3&\le \int _{{\mathbb {R}}^d} \left| f(z + e_T) - f(z) \right| p_{T,p,D}(z)dz\\&\le \sum _{|\alpha | = 1} \int _{{\mathbb {R}}^d} \left| \partial ^{\alpha } f(z + \tau e_T) \right| |e_T| p_{T,p,D}(z)dz \\&\lesssim |e_T| \left( \sup _{\begin{array}{c} |\alpha | \le \varGamma \\ x \in {\mathbb {R}}^d \end{array}}\frac{\left| \partial ^\alpha f(x) \right| }{1 + |x|^{L_1}}\right) \int _{{\mathbb {R}}^d} (1 + |z + e_T|^{L_1}) p_{T,p,D}(z)dz\\&\lesssim T^{-L_4 + \frac{L_5\varepsilon }{2} } \lesssim T^{-\frac{p-2}{2}}. \end{aligned}$$

Therefore, we get the conclusion. $\square$

1.2 Proofs of Section 2.2

Before prove Proposition 2.2, we consider an asymptotic expansion of $\tilde{Z}_T$. We assume that $Z_T$ satisfies the conditions [A1] and [A2]. Then, from Theorem 2.1, for any $L_1, L_2>0$, there exist $D>0$ and $\varGamma \in {\mathbb {N}}$ such that for any $f \in {\mathscr {E}}(\varGamma , L_1, L_2)$,

$$\begin{aligned} \left| E\left[ f\left( \frac{Z_T}{\sqrt{T}}\right) \right] -\int _{{\mathbb {R}}^d}f(z) p_{T,3,D}(z)dz \right| = o\left( T^{-1/2}\right) , \end{aligned}$$

where

$$\begin{aligned} p_{T,3,D}(z) = \phi (z;\varSigma _{T,D}) + \frac{1}{6\sqrt{T}}\kappa ^{a_1a_2a_3;}_T h_{a_1a_2a_3}(z;\varSigma _{T,D})\phi (z;\varSigma _{T,D}), \end{aligned}$$

for the modified cumulant $\kappa ^{a_1a_2a_3;}_T$ and the Hermite polynomial $h_{a_1a_2a_3}(z;\varSigma _{T,D})$ defined in (2.6) and (2.7) respectively. From the concrete form of $C_T$ and $M_{T, D}$, they are clearly non-degenerate and $f \circ M_{T, D} \circ C_T \in {\mathscr {E}}(\varGamma , L_1, L_2)$ holds for any $f \in {\mathscr {E}}(\varGamma , L_1, L_2)$. Owing to the variable transformation and the multi-linearity of the cumulant, the following inequality is immediately obtained. (See Proposition 7.1 in Sakamoto and Yoshida 2004 for the proof details.)

Lemma 6.5

Let $L_1, L_2>0$. Suppose that the conditions [A1]–[A3] and [B0] hold. Then, there exist $D>0$ and $\varGamma \in {\mathbb {N}}$ such that for any $f \in {\mathscr {E}}(\varGamma ,, L_1, L_2)$,

$$\begin{aligned} \left| E\left[ f\big (\tilde{Z}_T\big )\right] - \int _{{\mathbb {R}}^d}f(z) \tilde{p}_{T,3,D}(z)dz \right| = o\left( T^{-1/2}\right) , \end{aligned}$$

where $\tilde{\lambda }^{a_1a_2a_3;}_T$ is the $(a_1, a_2, a_3)$-cumulant of $\tilde{Z}_T$, $\tilde{\kappa }^{a_1a_2a_3;}_T =T^{1/2}\tilde{\lambda }^{a_1a_2a_3;}_T$ and

$$\begin{aligned} \tilde{p}_{T,3,D}(z) = \phi (z;\tilde{\varSigma }_{T,D}) +\frac{1}{6\sqrt{T}}\tilde{\kappa }^{a_1a_2a_3;}_T h_{a_1a_2a_3}(z;\tilde{\varSigma }_{T,D})\phi (z;\tilde{\varSigma }_{T,D}). \end{aligned}$$

Proof

(Proof of Proposition 2.2) It is proved in the same way as the proof of Theorem 5.1 in Sakamoto and Yoshida (2004) for

$$\begin{aligned} \tilde{q}_{T,3,D}(z^{(1)})&= \int _{{\mathbb {R}}^{p^2}}\phi (z;\tilde{\varSigma }_{T,D})dz^{(2)} + \frac{1}{\sqrt{T}}\left\{ \int _{{\mathbb {R}}^{p^2}}\frac{1}{6}\tilde{\kappa }^{a_1a_2a_3;}_T h_{a_1a_2a_3}(z;\tilde{\varSigma }_{T,D})\phi (z;\tilde{\varSigma }_{T,D})dz^{(2)} \right. \\&\quad \left. - \sum _{a=1,\dots ,p} \frac{\partial }{\partial z^{a;}} \int _{{\mathbb {R}}^{p^2}} \tilde{Q}_1^{a;}(z)\phi (z;\tilde{\varSigma }_{T,D})dz^{(2)} \right\} . \end{aligned}$$

by using the Bhattacharya–Ghosh map and transforming asymptotic expansion in Lemma 6.5. Thus, We only need to consider the form of $\tilde{q}_{T,3,D}$. Due to the orthogonalization, it immediately follows that

$$\begin{aligned} h_A(z;\tilde{\varSigma }_{T,D})\phi (z;\tilde{\varSigma }_{T,D}) =h_{A^{(1)}}(z^{(1)}; \tilde{\varSigma }_{T,D}^{(1,1)})\phi (z^{(1)}; \tilde{\varSigma }_{T,D}^{(1,1)})h_{A^{(2)}}(z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)}) \phi (z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)}) \end{aligned}$$

for $A^{(1)} = A\cap \{1,\dots ,p\}$ and $A^{(2)} = A\cap \{p+1,\dots ,p+p^2\}$. We decompose the polynomial $\tilde{Q}_1^{a;}(z)$ by the Hermite polynomials. Let

$$\begin{aligned} \tilde{Q}_1^{a;}(z) = \pi ^{a;}_{1,\phi }(z^{(1)}) + \pi ^{a;}_{1,a_1} (z^{(1)})h^{a_1;}(z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)}), \end{aligned}$$

where $h^{a;}(x;\sigma ) = \sigma ^{aa_1;}h_{a_1}(x;\sigma )$ for $\sigma = (\sigma ^{ab;})_{a,b=1, \dots , p}$ and $\pi ^{a;}_{1,\phi }(z^{(1)}), \pi ^{a;}_{1,a_1}(z^{(1)})$ are polynomials for $z^{(1)}$. It is well known that the orthogonality of the Hermite polynomial

$$\begin{aligned} \int h^A(z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)}) h_B(z^{(2)}; \tilde{\varSigma }_{T,D}^{(2, 2)}) \phi (z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)}) dz^{(2)} = {\left\{ \begin{array}{ll} A! &{} \text {if } A=B \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

This orthogonality gives the following representation of $\pi ^{a;}_{1,\phi }(z^{(1)})$,

$$\begin{aligned} \pi ^{a;}_{1,\phi }(z^{(1)})= & {} \int _{{\mathbb {R}}^{p^2}} \tilde{Q}_1^{a;}(z) \phi (z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)}) dz^{(2)} =\tilde{\mu }^{a;}_{a_1a_2}z^{a_1;}z^{a_2;}\\= & {} \tilde{\mu }^{a;}_{a_1a_2}\tilde{g}^{a_1b_1;} \tilde{g}^{a_2b_2;}h_{b_1b_2}(z^{(1)};\tilde{\varSigma }_{T,D}^{(1,1)}) + \tilde{\mu }^{a;}_{a_1a_2}\tilde{g}^{a_1a_2;}, \end{aligned}$$

where we used $h_{b_1b_2}(z^{(1)};\tilde{\varSigma }_{T,D}^{(1,1)}) =\tilde{g}_{b_1a_1}\tilde{g}_{b_2a_2}z^{a_1;}z^{a_2;} -\tilde{g}_{b_1b_2}$. Therefore, we get

$$\begin{aligned}&\sum _{a=1,\dots ,p} \frac{\partial }{\partial z^{a;}} \int _{{\mathbb {R}}^{p^2}}\tilde{Q}_1^{a;}(z)\phi (z;\tilde{\varSigma }_{T,D})dz^{(2)}\\&\quad = \sum _{a=1,\dots ,p} \frac{\partial }{\partial z^{a;}} \int _{{\mathbb {R}}^{p^2}} \left( \pi ^{a;}_{1,\phi }(z^{(1)}) + \pi ^{a;}_{1,a_1} (z^{(1)})h^{a_1;}(z^{(2)};\tilde{\varSigma }_{T,D}^{(2, 2)})\right) \\&\qquad \phi (z^{(1)};\tilde{\varSigma }_{T,D}^{(1,1)})\phi (z^{(2)}; \tilde{\varSigma }_{T,D}^{(2, 2)})dz^{(2)}\\&\quad = -\left( \tilde{\mu }^{a;}_{a_1a_2}\tilde{g}^{a_1b_1;} \tilde{g}^{a_2b_2;}h_{b_1b_2a}(z^{(1)};\tilde{\varSigma }_{T,D}^{(1,1)}) +\tilde{\mu }^{a;}_{a_1a_2}\tilde{g}^{a_1a_2;}h_a(z^{(1)}; \tilde{\varSigma }_{T,D}^{(1,1)})\right) \phi (z^{(1)};\tilde{\varSigma }_{T,D}^{(1,1)}). \end{aligned}$$

Thus, we can get the desired form of $\tilde{q}_{T,3,D}$. $\square$

Proof

(Proof of Proposition 2.3) This proof is the almost same as Theorem 6.2. in Sakamoto and Yoshida (2004). Let $\gamma ' \in \left( \frac{2}{3}, \gamma - \frac{L}{q_2} \right)$ and $\gamma '' \in \left( \frac{L}{q_3}, 3\gamma - 2 \right)$. We set

$$\begin{aligned} {\mathscr {X}}_{T,0}= & {} \left\{ \omega \in \varOmega \Bigg | \inf _{\begin{array}{c} T>0, |x|=1\\ \theta _1, \theta _2 \in \tilde{\varTheta } \end{array}} \left| x' \int ^1_0 \nu _{a b}\left( \theta _1 + s(\theta _2 - \theta _1)\right) ds\right| \right. \\> & {} \left. C', \ \left| T^{-\frac{2-\gamma }{2}}l_{a_1} \right|< C', \ \sup _{\theta \in \varTheta }\left| T^{-1}l_{a_1a_2}(\theta ) - \nu _{a_1a_2}(\theta ) \right| < \frac{C'}{2p^2} \right\} \end{aligned}$$

for some constant $C'>0$, and

$$\begin{aligned} {\mathscr {X}}_{T,1}= & {} \Bigg \{ \omega \in \varOmega \ \Bigg | \ \left| T^{-1}l_{a_1a_2} - \nu _{a_1a_2} \right|< T^{-\frac{\gamma '}{2}}, \ \left| T^{-1}l_{a_1a_2a_3} - \nu _{a_1a_2a_3} \right| \\< & {} T^{-\frac{\gamma '}{2}}, \ \sup _{\theta \in \varTheta }\left| T^{-1}l_{a_1a_2a_3a_4}(\theta )\right| < T^{\frac{\gamma ''}{2}} \Bigg \}. \end{aligned}$$

For appropriate $C'>0$ and sufficiently large T, it is known that there exists a unique $\hat{\theta }_T\in \tilde{\varTheta }$ such that $\partial _{\theta }l_T(\hat{\theta }_T) = 0$ and $|\hat{\theta }_T- \theta _0| <T^{-\frac{\gamma }{2}}$ on the set ${\mathscr {X}}_{T,0}$. In particular, ${\mathscr {X}}_{T,0} \subset \varOmega _T$ holds for large T. Moreover, it is also proved that $P[({\mathscr {X}}_{T,0})^c] \lesssim \ T^{-\frac{L}{2}}$. Here, we used the conditions [B0], [B1], [B2] and [B3], see the proof of Theorem 6.1 in Sakamoto and Yoshida (2004) for details. On the other hand, the conditions [B2] and [B4] lead

$$\begin{aligned} P[({\mathscr {X}}_{T,1})^c]&\le P\left[ T^{\frac{\gamma }{2}}\left| T^{-1}l_{a_1a_2} - \nu _{a_1a_2} \right| \ge T^{-\frac{\gamma '}{2} + \frac{\gamma }{2}}\right] \\&\quad + P\left[ T^{\frac{\gamma }{2}}\left| T^{-1}l_{a_1a_2a_3} - \nu _{a_1a_2a_3} \right| \ge T^{-\frac{\gamma '}{2} + \frac{\gamma }{2}}\right] \\&\quad + P\left[ \sup _{\theta \in \varTheta }\left| T^{-1}l_{a_1a_2a_3a_4}(\theta )\right| \ge T^{\frac{\gamma ''}{2}} \right] \\&\lesssim T^{-\frac{(\gamma - \gamma ')q_2}{2}} + T^{-\frac{\gamma '' q_3}{2}} \lesssim T^{-\frac{L}{2}}. \end{aligned}$$

Since $g_T^{-1}$ converge to a non-singular matrix by the condition [A3], we have $|Z^{a;}| \lesssim T^{-\frac{\gamma -1}{2}}$, $|Z^{a;}_{a_1}| \lesssim T^{-\frac{\gamma ' -1}{2}}$, $|Z^{a;}_{a_1a_2}| \lesssim T^{-\frac{\gamma ' -1}{2}}$ and $|\bar{\theta }| \lesssim T^{-\frac{\gamma -1}{2}}$ on ${\mathscr {X}}_{T,0} \cap {\mathscr {X}}_{T,1}$. Moreover, the condition [B4] guarantees

$$\begin{aligned} \big | \nu ^{a;}_{a_1 a_2} \big | \le \big | g^{ab;} \big | E \left[ \left| T^{-1}l_{b,a_1 a_2}\right| \right] < \infty \quad \text {uniformly with respect to}\ T. \end{aligned}$$

(6.6)

Hereafter, we consider the following inequalities on ${\mathscr {X}}_{T,0} \cap {\mathscr {X}}_{T,1}$. Let $a \in \{1, \dots , p\}$. First, we get

$$\begin{aligned} \left| T^{-\frac{1}{2}} \bar{R}^{a;}_2 \right|= & {} \Bigg |T^{-\frac{1}{2}} \left( \frac{1}{2}Z^{a;}_{a_1 a_2}\bar{\theta }^{a_1 a_2;} \right. \\&\left. +\frac{1}{2} \left\{ \int ^1_0(1-u)^2g^{ab;}\left( \frac{1}{T}l_{b a_1 a_2 a_3} \left( \theta _0 + u(\hat{\theta }_T-\theta _0)\right) \right) du\right\} \bar{\theta }^{a_1 a_2 a_3;} \right) \Bigg | \\\lesssim & {} T^{-\frac{1}{2} -\frac{\gamma ' -1}{2} - (\gamma -1)} + T^{-\frac{1}{2} +\frac{\gamma ''}{2} - \frac{3(\gamma -1)}{2}} \lesssim T^{-\frac{\varepsilon }{2}} \end{aligned}$$

for some small constant $0< \varepsilon < \min (2\gamma + \gamma ' -2, 3\gamma - \gamma '' -2)$. Similarly, we have

$$\begin{aligned} \left| \bar{R}^{a;}_1 \right| = \left| Z^{a;}_{a_1}\bar{\theta }^{a_1;} + \frac{1}{2}\nu ^{a;}_{a_1 a_2}\bar{\theta }^{a_1 a_2;} + T^{-\frac{1}{2}}\bar{R}^{a;}_2 \right| \lesssim T^{ -\frac{\gamma ' -1}{2} -\frac{\gamma -1}{2} } + T^{- (\gamma -1)} + T^{-\frac{\varepsilon }{2}} \lesssim T^{-\frac{\varepsilon '}{2}} \end{aligned}$$

for a positive constant $0< \varepsilon ' < \min (\gamma + \gamma ' -2, \varepsilon )$. Finally we have

$$\begin{aligned} \left| T^{-\frac{1}{2}} \check{R}^{a;}_2 \right|= & {} \left| T^{-\frac{1}{2}} \left( Z^{a;}_{a_1}\bar{R}^{a_1; }_1 +\bar{R}^{a;}_2 \right) + T^{-1}\left( \frac{1}{2}\nu ^{a;}_{a_1 a_2}\bar{R}^{a_1; }_1\bar{R}^{a_2; }_1 \right) \right| \\\lesssim & {} T^{-\frac{1}{2} -\frac{\gamma ' -1}{2} -\frac{\varepsilon '}{2}} +T^{-\frac{\varepsilon }{2}} + T^{-1 -\varepsilon '} \lesssim T^{-\frac{\varepsilon '}{2}}. \end{aligned}$$

Therefore, we get the desired conclusion

$$\begin{aligned} P\left[ \varOmega _T \cap \left\{ T^{-1}|\check{R}_2^{a;}| \le CT^{-\frac{1 + \varepsilon '}{2}}, \ a = 1, \dots , p \right\} \right] \ \ge \ P[{\mathscr {X}}_{T,0} \cap {\mathscr {X}}_{T,1} ] = 1 - o(T^{-\frac{L}{2}}). \end{aligned}$$

$\square$

Proof

(Proof of Theorem 2.4) From Proposition 2.2, we see that

From the definition of $\tilde{S}_T$ and the representation of (2.13),

$$\begin{aligned} \big \Vert \tilde{S}_T \big \Vert _{L^k(P)} \le \sum _{a=1,\dots , p} \bigg \Vert Z^{a;} +T^{-\frac{1}{2}}Z^{a;}_{a_1}Z^{a_1; } + \frac{1}{2}T^{-\frac{1}{2}} \nu ^{a;}_{a_1 a_2}Z^{a_1; }Z^{a_2; } \bigg \Vert _{L^k(P)}. \end{aligned}$$

By Corollary 6.3 and the representation of moments by cumulants, we have

$$\begin{aligned} \big \Vert T^{-\frac{1}{2}}Z_T \big \Vert _{L^k(P)} \lesssim T^{\varepsilon k} \end{aligned}$$

for any $\varepsilon > 0$ and $k>0$. From (6.6) and the above inequality, $\big \Vert \tilde{S}_T \big \Vert _{L^k(P)} \lesssim T^{\varepsilon k}$ holds for any $\varepsilon > 0$ and $k>0$. Thus, the condition [C1] yields $\big \Vert T^{-1}\check{R}_2 \big \Vert _{L^k(P)} \lesssim T^{\varepsilon k}$ for any $\varepsilon > 0$ and $k>0$.

We evaluate $\varDelta _1, \varDelta _2, \varDelta _3$ and $\varDelta _4$. From Proposition 2.3 and by choosing sufficiently small $\varepsilon$, we get

$$\begin{aligned} \varDelta _1 \lesssim E\left[ \left( 1 + \big |\tilde{S}_T\big | +\big |\sqrt{T}( \hat{\theta }_T- \theta _0 )\big | \right) ^{L_1} 1_{\varOmega _T^c}\right] \lesssim T^{-\frac{1}{2}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \varDelta _2 \lesssim E\left[ \left( 1 + \big |\tilde{S}_T\big | +\big |T^{-1}\check{R}_2\big | \right) ^{L_1} 1_{\varOmega _T \cap \left\{ T^{-1}|\check{R}_2^{a;}| > CT^{-\frac{1 + \varepsilon '}{2}}\right\} }\right] \lesssim T^{-\frac{1}{2}}. \end{aligned}$$

On the other hand, from the Taylor expansion, we have

$$\begin{aligned} f\big ( \tilde{S}_T + T^{-1}\check{R}_2\big ) - f\big ( \tilde{S}_T\big )= & {} \sum _{|\alpha |=1}T^{-1}\check{R}_2\int _0^1\partial ^{\alpha }f\big ( \tilde{S}_T + uT^{-1}\check{R}_2\big )du \\\lesssim & {} T^{-1}\big |\check{R}_2\big |\left( 1 + \big |\tilde{S}_T\big | +\big |T^{-1}\check{R}_2\big | \right) ^{L_1}. \end{aligned}$$

Thus, we get

$$\begin{aligned} \varDelta _3 \lesssim E\left[ T^{-1}\big |\check{R}_2\big |\left( 1 + \big |\tilde{S}_T\big | + \big |T^{-1}\check{R}_2\big | \right) ^{L_1} 1_{\left\{ T^{-1}|\check{R}_2^{a;}| \le CT^{-\frac{1 + \varepsilon '}{2}}\right\} }\right] \lesssim T^{-\frac{1}{2} - \frac{\varepsilon '}{2} + \varepsilon L_1} \lesssim T^{-\frac{1}{2}}, \end{aligned}$$

since we can choose small $\varepsilon$ arbitrary. Finally, we only have to show that $\varDelta _4 \lesssim T^{-\frac{1}{2}}$. From the definition of f,

$$\begin{aligned} \varDelta _4\lesssim & {} \int _{{\mathbb {R}}^d} (1+|z^{(1)}|)^{L1} \left| \tilde{q}_{T,3,D}(z^{(1)}) - q_{T,3}(z^{(1)}) \right| dz^{(1)} \nonumber \\\lesssim & {} \int _{{\mathbb {R}}^d} (1+|z^{(1)}|)^{L1} \left| \frac{\tilde{q}_{T,3,D}(z^{(1)})}{\phi (z^{(1)}; \tilde{g}_T^{-1})} -\frac{q_{T,3}(z^{(1)})}{\phi (z^{(1)}; g_T^{-1})} \right| \phi (z^{(1)}; \tilde{g}_T^{-1}) dz^{(1)} \nonumber \\&+ \int _{{\mathbb {R}}^d} (1+|z^{(1)}|)^{L1} \left| \frac{q_{T,3}(z^{(1)})}{\phi (z^{(1)}; g_T^{-1})} \left( 1 -\frac{\phi (z^{(1)}; g_T^{-1})}{\phi (z^{(1)}; \tilde{g}_T^{-1})} \right) \right| \phi (z^{(1)}; \tilde{g}_T^{-1}) dz^{(1)}.\qquad \end{aligned}$$

(6.7)

We see that $\tilde{g}_T - g_T = ( I - g_T\tilde{g}_T^{-1})\tilde{g}_T = T^{-D}g_T^{-1}\tilde{g}_T$. Since $(g_T^{-1}\tilde{g}_T)^{-1} = I + T^{-D}g_T^{-1}$ is positive definite, $g_T^{-1}\tilde{g}_T$ is also positive definite. With the help of the conditions [A2]–[A3] and Corollary 6.3, we can choose a sufficiently large $K>0$ such that

$$\begin{aligned}&\left| \frac{\tilde{q}_{T,3,D}(z^{(1)})}{\phi (z^{(1)}; \tilde{g}_T^{-1})} - \frac{q_{T,3}(z^{(1)})}{\phi (z^{(1)}; g_T^{-1})} \right| \nonumber \\&\quad = \frac{1}{\sqrt{T}}\Bigg | \Bigg \{ \left( \frac{1}{6}\tilde{\kappa }^{a_1a_2a_3;}_T + \tilde{\mu }^{a_3;}_{b_1b_2}\tilde{g}^{b_1a_1;}\tilde{g}^{b_2a_2;}\right) h_{a_1a_2a_3}(z^{(1)}; \tilde{g}_T^{-1}) + \tilde{\mu }^{a_1;}_{b_1b_2}\tilde{g}^{b_1b_2;}h_{a_1}(z^{(1)}; \tilde{g}_T^{-1}) \Bigg \} \nonumber \\&\qquad - \Bigg \{ \left( \frac{1}{6}\tilde{\kappa }^{a_1a_2a_3;}_T + \mu ^{a_3;}_{b_1b_2}g^{b_1a_1;}g^{b_2a_2;}\right) h_{a_1a_2a_3}(z^{(1)}; g_T^{-1}) + \mu ^{a_1;}_{b_1b_2}g^{b_1b_2;}h_{a_1}(z^{(1)}; g_T^{-1}) \Bigg \}\Bigg | \nonumber \\&\quad \lesssim T^{-D}(1 + |z^{(1)}| )^K, \end{aligned}$$

(6.8)

and

$$\begin{aligned} \left| \frac{q_{T,3}(z^{(1)})}{\phi (z^{(1)}; g_T^{-1})} \right| \lesssim (1 + |z^{(1)}| )^K. \end{aligned}$$

(6.9)

On the other hand, we obtain

$$\begin{aligned} \left| 1 - \frac{\phi (z^{(1)}; g_T^{-1})}{\phi (z^{(1)}; \tilde{g}_T^{-1})} \right|&= \left| 1 -\sqrt{\frac{|\tilde{g}_T^{-1}|}{|g_T^{-1}|}} \exp \left( -\frac{1}{2} z^{(1)'} \left( g_T - \tilde{g}_T \right) z^{(1)} \right) \right| \nonumber \\&\le \left| 1 - \sqrt{\frac{|\tilde{g}_T^{-1}|}{|g_T^{-1}|}} \right| + \sqrt{\frac{|\tilde{g}_T^{-1}|}{|g_T^{-1}|}} \left| 1 - \exp \left( -\frac{T^{-D}}{2} z^{(1)'} g_T^{-1}\tilde{g}_T z^{(1)} \right) \right| \nonumber \\&\le \left| 1 - \sqrt{\frac{|g_T^{-1} + T^{-D}(g_T^{-1})^2|}{|g_T^{-1}|}}\right| \nonumber \\&\quad + \sqrt{\frac{|g_T^{-1} + T^{-D}(g_T^{-1})^2|}{|g_T^{-1}|}}T^{-D} |z^{(1)'} g_T^{-1}\tilde{g}_T z^{(1)}| \nonumber \\&\lesssim T^{-\frac{D}{2}} +T^{-D} |z^{(1)}|^2. \end{aligned}$$

(6.10)

From (6.7), (6.8), (6.9) and (6.10), we get the conclusion by taking sufficiently large $D>0$. $\square$

1.3 Proofs of Section 3.2

Throughout this subsection, denote the i-th jump time of $N^x_t$ by $\tau ^x_i$.

Proof

(Proof of Proposition 3.5) Let $M_1, K_1$ and $K_2$ be positive constants and ${\mathscr {A}}$ be the operator in Lemma 3.4. Define g(y, t) and $\bar{{\mathscr {A}}}$ by $g(y, t) = e^{M_1y}e^{K_1t}$ and $\bar{{\mathscr {A}}}g(y, t) = e^{K_1t}({\mathscr {A}}e^{M_1y} +K_1e^{M_1y})$. From Lemma 3.4,

$$\begin{aligned} \bar{{\mathscr {A}}}g(y, t) \le e^{K_1t}(-K_1e^{M_1y} + K_2 + K_1e^{M_1y}) = e^{K_1t}K_2. \end{aligned}$$

(6.11)

Since $\lambda ^x_{t \wedge \tau ^x_i}$ is bounded, thus $g(\lambda ^x_{t \wedge \tau ^x_i}, t \wedge \tau ^x_i) =e^{M_1 \lambda ^x_{t \wedge \tau ^x_i}}e^{K_1(t \wedge \tau ^x_i)}$ is integrable. Furthermore, one may get

$$\begin{aligned} g(\lambda ^x_{t \wedge \tau ^x_i}, t \wedge \tau ^x_i) - g(\lambda ^x_0, 0)&=\int _{(0,t \wedge \tau ^x_i]} g(\lambda ^x_s + \alpha , s) - g(\lambda ^x_s, s) dN^x_s \nonumber \\&\quad + \int _{(0,t \wedge \tau ^x_i]} \frac{d}{ds} g(\lambda ^x_s, s) ds \nonumber \\&= \int _{(0, t \wedge \tau ^x_i]} g(\lambda ^x_s + \alpha , s) -g(\lambda ^x_s, s) d\tilde{N}^x_s\nonumber \\&\quad + \int _{(0, t \wedge \tau ^x_i]} \bar{{\mathscr {A}}}g(\lambda ^x_s, s)ds. \end{aligned}$$

(6.12)

Since $\int _{(0, t]} g(\lambda ^x_s + \alpha , s) - g(\lambda ^x_s, s) d\tilde{N}^x_s$ is a $\tau ^x_i$-local martingale, see Theorem 18.7 in Liptser and Shiryaev (2000), (6.11) and (6.12) yield

$$\begin{aligned} E\left[ g(\lambda ^x_{t\wedge \tau ^x_i}, t \wedge \tau ^x_i) \right]= & {} g(x, 0) + E\left[ \int _{(0, t \wedge \tau ^x_i]} \bar{{\mathscr {A}}} g(\lambda ^x_s, s) ds\right] \\\le & {} g(x, 0) + E\left[ \int _{(0, t \wedge \tau ^x_i]} e^{K_1t}K_2 ds\right] \le g(x, 0) + \frac{K_2}{K_1}\left( e^{K_1t} - 1\right) . \end{aligned}$$

Then, by the Fatou’s lemma, we have

$$\begin{aligned} E\left[ e^{M_1\lambda ^x_t}\right] \le e^{-K_1t}\left( g(x, 0) +\frac{K_2}{K_1}\left( e^{K_1t} - 1\right) \right) . \end{aligned}$$

Thus, we get the conclusion. $\square$

Proof

(Proof of Lemma 3.7) Since ${\mathscr {A}}$ is linear, we only have to prove that, for $p(y) = y^m$ with $m \in {\mathbb {N}}$, $M^p_t = p(\lambda ^x_t) - p(\lambda ^x_0) - \int _{(0,t]} {\mathscr {A}}p(\lambda ^x_s) ds$ is a ${\mathscr {F}}^x_t$-martingale. Take any large $T>0$. In the same way as (6.12) in the proof of Proposition 3.5, one may confirm that $M^p_t = \int _0^{t} 1_{\{s\le T\}}(\lambda ^x_s + \alpha )^m - (\lambda ^x_s)^m d\tilde{N}^x_s$ for $t \le T$. Then, Theorem 18.7 in Liptser and Shiryaev (2000) and Proposition 3.5 lead the conclusion. $\square$

Proof

(Proof of Lemma 3.8) Let $p(y) = a_my^m + \cdots a_1y + a_0$, where $a_0, \dots , a_m \in {\mathbb {R}}$ and $m \in {\mathbb {N}}$. Then, the linearity of ${\mathscr {A}}$ leads

$$\begin{aligned}&E\left[ \left| \int _{(s,t]}\int _{(s,u_1]}\cdots \int _{(s,u_{n-1}]}E \left[ \left. {\mathscr {A}}^np(\lambda ^x_{u_n})\right| {\mathscr {F}}^x_s\right] du_n\dots du_2du_1\right| \right] \\&\quad \le \int _{(s,t]}\int _{(s,u_1]}\cdots \int _{(s,u_{n-1}]}E \left[ \left| {\mathscr {A}}^np(\lambda ^x_{u_n})\right| \right] du_n\dots du_2du_1\\&\quad \le \sum _{k=1}^m |a_k|\int _{(s,t]}\int _{(s,u_1]} \cdots \int _{(s,u_{n-1}]}E\left[ \left| {\mathscr {A}}^n(\lambda ^x_{u_n})^k\right| \right] du_n\dots du_2du_1. \end{aligned}$$

Therefore, we only have to evaluate $\int _{(s,t]}\int _{(s,u_1]}\cdots \int _{(s,u_{n-1}]}E\left[ \left| {\mathscr {A}}^n(\lambda ^x_{u_n})^k\right| \right] du_n\dots du_2du_1$ for any $k \in {\mathbb {N}}$. In particular,

$$\begin{aligned} \int _{(s,t]}\int _{(s,u_1]}\cdots \int _{(s,u_{n-1}]}E\left[ \left| {\mathscr {A}}^n(\lambda ^x_{u_n})^k\right| \right] du_n\dots du_2du_1 \le \frac{(t-s)^n}{n!}\sup _{u\in (s,t]}E\left[ \left| {\mathscr {A}}^n(\lambda ^x_u)^k\right| \right] , \end{aligned}$$

thus it is enough to prove that the above right hand side converges to zero as $n \rightarrow \infty$. There exist constants $C_i$, $i=0,\dots , k$ such that ${\mathscr {A}}y^k = C_ky^k + \cdots + C_1y + C_0$. Then, we inductively get

$$\begin{aligned} {\mathscr {A}}^ny^k= & {} {\mathscr {A}}^{n-1}({\mathscr {A}}y^k) = C_k{\mathscr {A}}^{n-2}({\mathscr {A}}y^k) +\sum _{i=1}^{k-1}C_i{\mathscr {A}}^{n-1}y^i \nonumber \\= & {} C_k^2{\mathscr {A}}^{n-3}({\mathscr {A}}y^k) +C_k\sum _{i=1}^{k-1}C_i{\mathscr {A}}^{n-2}y^i +\sum _{i=1}^{k-1}C_i{\mathscr {A}}^{n-1}y^i \nonumber \\= & {} \cdots = C_k^{n-1}{\mathscr {A}}y^k +C_k^{n-2}\sum _{i=1}^{k-1}C_i{\mathscr {A}}y^i +C_k^{n-3}\sum _{i=1}^{k-1}C_i{\mathscr {A}}^2y^i + \cdots +\sum _{i=1}^{k-1}C_i{\mathscr {A}}^{n-1}y^i \nonumber \\= & {} C_k^ny^k +\sum _{i=1}^{k-1} C_i\left( C_k^{n-1}y^i +C_k^{n-2}{\mathscr {A}}y^i + \cdots + {\mathscr {A}}^{n-1}y^i\right) +C_0C_k^{n-1}. \end{aligned}$$

(6.13)

Hence,

$$\begin{aligned} \sup _{u\in (s,t]}E\left[ \left| {\mathscr {A}}^n(\lambda ^x_u)^k\right| \right]&\le C_k^n\sup _{u\in (s,t]}E\left[ \left| (\lambda ^x_u)^k\right| \right] \\&\quad + \sum _{i=1}^{k-1}C_i\sup _{u\in (s,t]}E\left[ \left| C_k^{n-1}(\lambda ^x_u)^i + C_k^{n-2}{\mathscr {A}}(\lambda ^x_u)^i + \cdots + {\mathscr {A}}^{n-1} (\lambda ^x_u)^i\right| \right] \\&\quad +C_0C_k^{n-1}. \end{aligned}$$

Furthermore, one may concretely compute as $C_k = k(\alpha -\beta )$. Now, we introduce the following assumption. $\square$

ASS(k) 1

For any $i= 0, \dots , k-1$ and $C = j(\alpha -\beta )$ with $j = k, k+1, \ldots$,

$$\begin{aligned} \frac{(t-s)^n}{n!}\sup _{u\in (s,t]}E\left[ \left| C^{n-1}(\lambda ^x_u)^i + C^{n-2}{\mathscr {A}}(\lambda ^x_u)^i + \cdots +{\mathscr {A}}^{n-1}(\lambda ^x_u)^i\right| \right] \rightarrow 0 \quad \text {as}\quad n \rightarrow \infty . \end{aligned}$$

Proposition 3.5 guarantees $\sup _{u\in (s,t]}E\left[ \left| (\lambda ^x_u)^k\right| \right] < \infty$. Thus, if ASS(k) holds, by taking $C = k(\alpha -\beta )$ in ASS(k), we have

$$\begin{aligned} \frac{(t-s)^n}{n!}\sup _{u\in (s,t]}E\left[ \left| {\mathscr {A}}^n (\lambda ^x_u)^k\right| \right] \rightarrow 0. \end{aligned}$$

We prove that ASS(k) holds for any $k \in {\mathbb {N}}$ by induction. In the case of $k=1$, this assumption is obvious. Assume that ASS(k) holds. Again we denote ${\mathscr {A}}y^k = C_ky^k + \cdots + C_1y +C_0$. For $C = j(\alpha -\beta )$ with $j = k+1, k+2, \ldots$, by using the Eq. (6.13), we have

$$\begin{aligned}&C^{n-1}y^k + C^{n-2}{\mathscr {A}}y^k + \cdots + {\mathscr {A}}^{n-1}y^k\\&\quad = C^{n-1}y^k + C^{n-2}\left\{ C_ky^k + \sum _{i=1}^{k-1}C_iy^i + C_0\right\} \\&\qquad + C^{n-3}\left\{ C_k^2y^k + \sum _{i=1}^{k-1}C_i(C_ky^{i} + {\mathscr {A}}y^{i}) + C_0C_k\right\} \\&\qquad + C^{n-4}\left\{ C_k^3y^k+ \sum _{i=1}^{k-1}C_i(C_k^2y^i +C_k{\mathscr {A}}y^i + {\mathscr {A}}^2y^i) + C_0C_k^2\right\} \\&\qquad +\cdots +\left\{ C_k^{n-1}y^k +\sum _{i=1}^{k-1} C_i\left( C_k^{n-2}y^i + C_k^{n-3}{\mathscr {A}}y^i + \cdots + {\mathscr {A}}^{n-2}y^i\right) +C_0C_k^{n-2}\right\} \\&\quad = \sum _{i=0}^{n-1}C^iC_k^{n-1-i}y^k+ C_{k-1}\sum _{j=0}^{n-2} \left( \sum _{i=0}^{n-2-j}C^iC_k^{n-2-i-j}\right) {\mathscr {A}}^{j}y^{k-1}\\&\qquad + C_{k-2}\sum _{j=0}^{n-2}\left( \sum _{i=0}^{n-2-j}C^iC_k^{n-2-i-j}\right) {\mathscr {A}}^{j}y^{k-2}\\&\qquad + \cdots + C_{1}\sum _{j=0}^{n-2}\left( \sum _{i=0}^{n-2-j}C^i C_k^{n-2-i-j}\right) {\mathscr {A}}^{j}y + C_0\left( \sum _{i=0}^{n-2}C^iC_k^{n-2-i}\right) \\&\quad = \frac{C^n-C_k^n}{C-C_k}y^k+ C_{k-1}\sum _{j=0}^{n-2}\frac{C^{n-1-j} -C_k^{n-1-j}}{C-C_k}{\mathscr {A}}^{j}y^{k-1} \\&\qquad + C_{k-2}\sum _{j=0}^{n-2} \frac{C^{n-1-j}-C_k^{n-1-j}}{C-C_k}{\mathscr {A}}^{j}y^{k-2}\\&\qquad + \cdots + C_{1}\sum _{j=0}^{n-2}\frac{C^{n-1-j}-C_k^{n-1-j}}{C-C_k} {\mathscr {A}}^{j}y + C_0\frac{C^{n-1}-C_k^{n-1}}{C-C_k}\\&\quad =\frac{1}{C-C_k}\left\{ (C^n-C_k^n)y^k +C_{k-1} \left\{ \sum _{i=0}^{n-1}C^i{\mathscr {A}}^{n-1-i}y^{k-1} -\sum _{i=0}^{n-1}C_k^i{\mathscr {A}}^{n-1-i}y^{k-1}\right\} \right. \\&\qquad \left. +C_{k-2}\left\{ \sum _{i=0}^{n-1}C^i{\mathscr {A}}^{n-1-i}y^{k-2} -\sum _{i=0}^{n-1}C_k^i{\mathscr {A}}^{n-1-i}y^{k-2}\right\} + \cdots \right. \\&\qquad \left. +C_1\left\{ \sum _{i=0}^{n-1}C^i{\mathscr {A}}^{n-1-i}y -\sum _{i=0}^{n-1}C_k^i{\mathscr {A}}^{n-1-i}y\right\} +C_0(C^{n-1}-C_k^{n-1})\right\} . \end{aligned}$$

Therefore,

$$\begin{aligned}&\frac{(t-s)^n}{n!}\sup _{u\in (s,t]}E\left[ \left| C^{n-1}(\lambda ^x_u)^k + C^{n-2}{\mathscr {A}}(\lambda ^x_u)^k + \cdots + {\mathscr {A}}^{n-1}(\lambda ^x_u)^k\right| \right] \\&\quad \le \frac{(t-s)^n}{n!}\left| \frac{1}{C-C_k}\right| \Bigg \{\left| C^n-C_k^n\right| \sup _{u\in (s,t]}E\left[ (\lambda ^x_u)^k\right] \\&\qquad +\left| C_{k-1}\right| \left\{ \sup _{u\in (s,t]}E \left[ \left| \sum _{i=0}^{n-1}C^i{\mathscr {A}}^{n-1-i} (\lambda ^x_u)^{k-1} \right| \right] +\sup _{u\in (s,t]}E \left[ \left| \sum _{i=0}^{n-1}C_k^i{\mathscr {A}}^{n-1-i} (\lambda ^x_u)^{k-1} \right| \right] \right\} \\&\qquad +\left| C_{k-2}\right| \left\{ \sup _{u\in (s,t]}E \left[ \left| \sum _{i=0}^{n-1}C^i{\mathscr {A}}^{n-1-i} (\lambda ^x_u)^{k-2} \right| \right] +\sup _{u\in (s,t]}E \left[ \left| \sum _{i=0}^{n-1}C_k^i{\mathscr {A}}^{n-1-i} (\lambda ^x_u)^{k-2} \right| \right] \right\} \\&\qquad +\cdots + \left| C_1\right| \left\{ \sup _{u\in (s,t]}E \left[ \left| \sum _{i=0}^{n-1}C^i{\mathscr {A}}^{n-1-i}\lambda ^x_u\right| \right] +\sup _{u\in (s,t]}E\left[ \left| \sum _{i=0}^{n-1}C_k^i{\mathscr {A}}^{n-1-i} \lambda ^x_u \right| \right] \right\} \\&\qquad +\left| C_0(C^{n-1}-C_k^{n-1})\right| \Bigg \}. \end{aligned}$$

Hence, ASS(k) leads ASS($\mathrm{k}+1$) and we have completed the proof. $\square$

To prove Theorem 3.9, we prepare the following lemmas.

Lemma 6.6

For any $u \in {\mathbb {R}}$ and $t \ge s \ge 0$, $E[e^{iu\lambda ^x_t}| {\mathscr {F}}^x_s] = E[e^{iu\lambda ^x_t}| \lambda ^x_s] \ \ a.s$.

Proof

Fix s and t with $t \ge s \ge 0$. It is sufficient to show that for any bounded ${\mathscr {F}}^x_s$-measurable function $g : \varOmega \rightarrow {\mathbb {R}}$,

$$\begin{aligned} E[e^{iu\lambda ^x_t}g] = E\left[ E[e^{iu\lambda ^x_t}| \lambda ^x_s]g\right] . \end{aligned}$$

Note that

$$\begin{aligned} E\left[ E[e^{iu\lambda ^x_t}| \lambda ^x_s]g\right] =E\left[ E[e^{iu\lambda ^x_t}| \lambda ^x_s]E[g| \lambda ^x_s]\right] =E\left[ e^{iu\lambda ^x_t}E[g| \lambda ^x_s]\right] . \end{aligned}$$

Let $D = \{z \in {\mathbb {C}}; \ {\text {Re}}z < \frac{M_1}{2}\}$, where $M_1$ is the positive constant chosen in Proposition 3.5. First, we will prove that $f(z) =E[e^{z\lambda ^x_t}g]$ is holomorphic on D for any ${\mathscr {F}}^x_s$-measurable function g. Let $z = a + ib$, where $a, b \in {\mathbb {R}}$ with $a < \frac{M_1}{2}$. Then, we have

$$\begin{aligned} \left| f(z)\right| \le \Vert g\Vert _{\infty }E[e^{a\lambda ^x_t}] < \infty , \end{aligned}$$

and thus $f(z)=E[e^{z\lambda ^x_t}g]$ is defined on D. Define u(a, b) and v(a, b) as the real part and the imaginary part of f(z) respectively, namely,

$$\begin{aligned} f(z) = E[\cos (b\lambda ^x_t)e^{a\lambda ^x_t}g] + iE[\sin (b\lambda ^x_t) e^{a\lambda ^x_t}g] = u(a,b) + iv(a,b). \end{aligned}$$

Write $\partial _a = \frac{\partial }{\partial a}$ and $\partial _b =\frac{\partial }{\partial b}$. $|\partial _a(\cos (b\lambda ^x_t) e^{a\lambda ^x_t}g)|$, $|\partial _b(\cos (b\lambda ^x_t) e^{a\lambda ^x_t}g)|$, $|\partial _a(\sin (b\lambda ^x_t) e^{a\lambda ^x_t}g)|$ and $|\partial _b(\sin (b\lambda ^x_t) e^{a\lambda ^x_t}g)|$ are dominated by an integrable random variable $|\lambda ^x_te^{\frac{M_1}{2}\lambda ^x_t}g|$ on D. Hence, the permutation of differential and integral is permitted, and thus we have

$$\begin{aligned} \partial _au(a,b)&= \partial _bv(a,b) = E[\lambda ^x_t\cos (b\lambda ^x_t) e^{a\lambda ^x_t}g] \quad \text { and } \\ \partial _bu(a,b)&= -\partial _a v(a,b) = -E[\lambda ^x_t\sin (b\lambda ^x_t)e^{a\lambda ^x_t}g]. \end{aligned}$$

The Lebesgue’s theorem guarantees that $\partial _au(a,b)$, $\partial _bu(a,b)$, $\partial _av(a,b)$ and $\partial _bv(a,b)$ are continuous with respect to a and b. In particular, they are total differentiable. Then, the Cauchy–Riemann relations lead that f(z) is holomorphic on D. Completely similarly, we can prove that $z \mapsto E\left[ e^{z\lambda ^x_t}E[g| \lambda ^x_s]\right]$ is also holomorphic on D.

Second, we will confirm that $E[e^{z\lambda ^x_t}g] =E\left[ e^{z\lambda ^x_t}E[g| \lambda ^x_s]\right]$ for $z \in (-\frac{M_1}{2}, \frac{M_1}{2})$. Let $p_N(x) =\sum _{n=0}^N\frac{(zx)^n}{n!}$. Then, (3.3) leads

$$\begin{aligned} E\left[ \sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!}g\right]= & {} E\left[ E\left[ \left. \sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!}g\right| {\mathscr {F}}^x_s\right] \right] = E\left[ E\left[ \left. \sum _{n=0}^N \frac{(z\lambda ^x_t)^n}{n!}\right| {\mathscr {F}}^x_s\right] g\right] \\= & {} E\left[ E\left[ \left. \sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!}\right| \lambda ^x_s\right] g\right] = E\left[ \sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!} E[g| \lambda ^x_s]\right] . \end{aligned}$$

On the other hand, since $\sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!} \rightarrow e^{z\lambda ^x_t}$ as $N \rightarrow \infty$ and $|\sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!}| \le |\sum _{n=0}^N\frac{(\frac{M_1}{2}\lambda ^x_t)^n}{n!}| \le e^{\frac{M_1}{2}\lambda ^x_t}$ hold for every $z \in (-\frac{M_1}{2}, \frac{M_1}{2})$, we have

$$\begin{aligned} \sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!}g \rightarrow e^{z\lambda ^x_t}g \quad \text {as}\quad N \rightarrow \infty \ \text {in}\ L^1\text {-sence}, \end{aligned}$$

and

$$\begin{aligned} \sum _{n=0}^N\frac{(z\lambda ^x_t)^n}{n!}E[g| \lambda ^x_s] \rightarrow e^{z\lambda ^x_t}E[g| \lambda ^x_s] \quad \text {as}\ N \rightarrow \infty \ \text {in}\ L^1\text {-sence} \end{aligned}$$

by the Lebesgue’s theorem. Therefore, we get the desired equation

$$\begin{aligned} E\left[ e^{z\lambda ^x_t}g\right] = E\left[ e^{z\lambda ^x_t} E[g| \lambda ^x_s]\right] \quad \text {for}\ z \in \left( -\frac{M_1}{2}, \frac{M_1}{2}\right) . \end{aligned}$$

Now, the identity theorem guarantees the conclusion. $\square$

Proof

(Proof of Theorem 3.9) For almost every $a, b \in {\mathbb {R}}$ with $a < b$, we will prove that

$$\begin{aligned} P \Big [\lambda ^x_t \in (a,b] \Big | {\mathscr {F}}^x_s \Big ] = \lim _{A \rightarrow \infty } \frac{1}{2\pi } \int _{-A}^A \frac{e^{-iua} -e^{-iub}}{iu}E\left[ e^{iu\lambda ^x_t} \Big | {\mathscr {F}}^x_s\right] du \quad a.s, \end{aligned}$$

(6.14)

and

$$\begin{aligned} P \Big [\lambda ^x_t \in (a,b] \Big | \lambda ^x_s \Big ] =\lim _{A \rightarrow \infty } \frac{1}{2\pi } \int _{-A}^A \frac{e^{-iua} -e^{-iub}}{iu}E\left[ e^{iu\lambda ^x_t} \Big | \lambda ^x_s\right] du \quad {\textit{a.s.}} \end{aligned}$$

(6.15)

However, by considering a probability measure $P_F(d\omega ) =P[d\omega \cap F]/P[F]$, the Lévy’s inversion formula gives

$$\begin{aligned} P \Big [ \big \{ \lambda ^x_t \in (a,b] \big \} \cap F \Big ] = \lim _{A \rightarrow \infty } \frac{1}{2\pi } \int _{-A}^A \frac{e^{-iua} -e^{-iub}}{iu}E\left[ e^{iu\lambda ^x_t} 1_F \right] du \end{aligned}$$

for any set $F \in {\mathscr {F}}^x_s$. Moreover, we know

$$\begin{aligned} \frac{1}{2\pi } \int _{-A}^A \frac{e^{-iua} - e^{-iub}}{iu}E \big [e^{iu\lambda ^x_t} 1_F \big ]du = E\left[ E\left[ \frac{1}{\pi } \big \{ D_A(\lambda ^x_t-a) - D_A(\lambda ^x_t-b) \big \} \Big | {\mathscr {F}}^x_s\right] 1_F \right] \end{aligned}$$

where $D_A$ is the Dirichlet integral, i.e.

$$\begin{aligned} D_A(\alpha ) = \int _0^A \frac{\sin (u\alpha )}{u}du. \end{aligned}$$

Then, as is well known, we can apply the Lebesgue’s theorem and get

$$\begin{aligned}&\lim _{A\rightarrow \infty } \frac{1}{2\pi } \int _{-A}^A \frac{e^{-iua} -e^{-iub}}{iu}E\left[ e^{iu\lambda ^x_t} 1_F \right] du\\&\quad = E\left[ \left( \lim _{A \rightarrow \infty } \frac{1}{2\pi } \int _{-A}^A \frac{e^{-iua} - e^{-iub}}{iu} E\left[ e^{iu\lambda ^x_t} \Big | {\mathscr {F}}^x_s\right] du \right) 1_F\right] . \end{aligned}$$

Thus, (6.14) holds. In the same way, (6.15) also holds. Then, from Lemma 6.6, we get $P[\lambda ^x_t \in (a,b] | {\mathscr {F}}^x_s] = P[\lambda ^x_t \in (a,b] | \lambda ^x_s] \ \ {\textit{a.s.}}$ for almost every $a, b \in {\mathbb {R}}$ with $a < b$. With the help of the monotone class theorem, $E[f(\lambda ^x_t)| {\mathscr {F}}^x_s] = E[f(\lambda ^x_t)| \lambda ^x_s] \ \ {\textit{a.s.}}$ holds for any bounded measurable function f. $\square$

1.4 Proofs of Section 3.3

1.4.1 Markovian property

Proof

(Proof of Proposition 3.11) From Theorem 3.9, we immediately get, for any $t \ge s \ge 0$ and bounded measurable function f,

$$\begin{aligned} E\big [f\big (X^{x,(1)}_t\big ) \big | {\mathscr {F}}^x_s \big ] = E\big [f\big (X^{x,(1)}_t\big ) \big | X^{x,(1)}_s \big ]\quad {{\textit{a.s.}}} \end{aligned}$$

(6.16)

$X_t^x$ has the following relation. For any $t \ge s \ge 0$,

$$\begin{aligned} X^{x,(2)}_t&= \left( x_1t + x_2\right) e^{-\beta t} + \int _{(0,s)} \alpha (t-u)e^{-\beta (t-u)}dN_u^{x_1} + \int _{[s,t)} \alpha (t-u)e^{-\beta (t-u)}dN_u^{x_1}\\&= \Bigg \{ (t-s)\Big (x_1e^{-\beta s} + \int _{(0,s)} \alpha e^{-\beta (s-u)}dN_u^{x_1}\Big )\\&\quad + \Big ( (x_1s + x_2) e^{-\beta s} + \int _{(0,s)} \alpha (s-u)e^{-\beta (s-u)}dN_u^{x_1} \Big ) \Bigg \} e^{-\beta (t-s)} \\&\quad + \int _{[s,t)} \alpha (t-u)e^{-\beta (t-u)}dN_u^{x_1}\\&= \left( X^{x,(1)}_s(t-s) + X^{x,(2)}_s \right) e^{-\beta (t-s)} +\int _{[s,t)} \alpha (t-u)e^{-\beta (t-u)}dN_u^{x_1}, \end{aligned}$$

and similarly,

$$\begin{aligned}&X^{x,(3)}_t = \left( X^{x,(1)}_s(t-s)^2 + 2X^{x,(2)}_s(t-s) + X^{x,(3)}_s\right) e^{-\beta (t-s)}\\&+ \int _{[s,t)} \alpha (t-u)^2e^{-\beta (t-u)}dN_u^{x_1}. \end{aligned}$$

Therefore, $X^x_t$ is $\sigma \big ( X^{x,(1)}_u, X^{x,(2)}_s, X^{x,(3)}_s; u \in [s,t] \big )$-measurable for any $t \ge s\ge 0$.

Let $g_1, g_2$ and $g_3$ be $\sigma \big ( X^{x,(1)}_u; u \in [s,t] \big ), \sigma \big ( X^{x,(2)}_s \big )$ and $\sigma \big (X^{x,(3)}_s \big )$ measurable bounded functions respectively. Then, (6.16) and the monotone class theorem lead

$$\begin{aligned} E[g_1|X^x_s] = E[g_1|{\mathscr {F}}^x_s|X^x_s] = E[g_1|X^{x,(1)}_s|X^x_s] = E[g_1|X^{x,(1)}_s] = E[g_1|{\mathscr {F}}^x_s]\quad {\textit{a.s.}} \end{aligned}$$

Thus,

$$\begin{aligned} E[g_1g_2g_3|{\mathscr {F}}^x_s] = g_2g_3E[g_1|{\mathscr {F}}^x_s] = g_2g_3E[g_1|X^x_s] = E[g_1g_2g_3|X^x_s]\quad {\textit{a.s.}} \end{aligned}$$

By the monotone class theorem, we get the conclusion. $\square$

Before we prove the homogeneous Markov property of process X, we prepare the following technical lemma.

Lemma 6.7

For any bounded function f defined on the path space of $X^{x,(1)}_u, u \in [s,t]$,

$$\begin{aligned} E\big [ f \big ( X^{x,(1)}_u; u \in [s,t] \big ) \big | {\mathscr {F}}^x_s \big ] (\omega ) = E\big [ f \big ( X^{y,(1)}_{u-s}; u \in [s,t] \big ) \big ] \big |_{y = X^{x,(1)}_s(\omega ) } \quad {\textit{a.s}}\quad \omega . \end{aligned}$$

Proof

From Theorem 3.9 and the monotone class theorem, $E\big [f \big ( X^{x,(1)}_u; u \in [s,t] \big ) \big | {\mathscr {F}}^x_s \big ] = E\big [f \big ( X^{x,(1)}_u; u \in [s,t] \big ) \big | X^{x,(1)}_s \big ] \ {\textit{a.s.}}$ holds. Therefore, we only have to prove the statement replaced ${\mathscr {F}}^x_s$ by $\sigma \big ( X^{x,(1)}_s \big )$. For any $p \in {\mathscr {P}}$, we know that for almost every $\omega \in \varOmega$ and $s \le t$,

$$\begin{aligned} E\left[ \left. p \big ( X^{x,(1)}_{t} \big ) \right| X^{x,(1)}_s\right] (\omega )= & {} E\left[ \left. p \left( \lambda ^{x_1}_{t} - \mu \right) \right| \lambda _s^{x_1} \right] (\omega ) \\= & {} e^{(t-s){\mathscr {A}}} p \left( \lambda ^{x_1}_{s}(\omega ) - \mu \right) \\= & {} e^{(t-s){\mathscr {A}}} p \left( \lambda ^{X^{x,(1)}_{s}(\omega )}_0 (\tilde{\omega })-\mu \right) \\= & {} E\left[ \left. p \left( \lambda ^y_{t-s}-\mu \right) \right] \right| _{y=X^{x,(1)}_s(\omega )}\\= & {} E\left[ \left. p \left( X^{y,(1)}_{t-s} \right) \right] \right| _{ y=X^{x,(1)}_s(\omega ) }, \end{aligned}$$

where the operator $e^{(t-s){\mathscr {A}}}$ is defined in Sect. 3.2. In particular, for $u \in {\mathbb {R}}$ with $|u| \le \frac{M_1}{2}$ and $p_N(y) = \sum _{n=0}^N \frac{(uy)^n}{n!} \in {\mathscr {P}}$,

$$\begin{aligned} E\left[ \left. p_N \big ( X^{x,(1)}_{t} \big ) \right| X^{x,(1)}_s\right] = E\left[ \left. p_N \left( X^{y,(1)}_{t-s} \right) \right] \right| _{ y=X^{x,(1)}_s } \quad {\textit{a.s.}} \end{aligned}$$

Moreover, Proposition 3.5 and the Lebesgue’s theorem give

$$\begin{aligned} E\left[ \left. e^{iuX^{x,(1)}_t } \right| X^{x,(1)}_s\right] =E\left[ \left. e^{iuX^{y,(1)}_{t-s}} \right] \right| _{ y=X^{x,(1)}_s } \quad {\textit{a.s.}} \end{aligned}$$

In the same way of the proof of Lemma 6.6, one may confirm that Proposition 3.5 and the identity theorem guarantee that for general $u \in {\mathbb {R}}$,

$$\begin{aligned} E\left[ \left. e^{iuX^{x,(1)}_t } \right| X^{x,(1)}_s\right] =E\left[ \left. e^{iuX^{y,(1)}_{t-s}} \right] \right| _{ y=X^{x,(1)}_s }\quad {\textit{a.s.}} \end{aligned}$$

Finally, it is proved in the same way as Theorem 3.9 that

$$\begin{aligned} E\left[ \left. f \big ( X^{x,(1)}_{t} \big ) \right| X^{x,(1)}_s\right] = E\left[ \left. f \left( X^{y,(1)}_{t-s} \right) \right] \right| _{ y=X^{x,(1)}_s } \quad {\textit{a.s.}} \end{aligned}$$

for any bounded measurable function f. Thus, for any $s \le u_1 \le u_2 \le t$ and bounded functions $g_1,g_2$,

$$\begin{aligned} E\bigg [ g_1\big ( X^{x,(1)}_{u_1} \big ) g_2\big ( X^{x,(1)}_{u_2} \big ) \bigg | X^{x,(1)}_s \bigg ]= & {} E\bigg [g_1\big ( X^{x,(1)}_{u_1} \big ) E\big [ g_2\big ( X^{x,(1)}_{u_2} \big ) \big | {\mathscr {F}}^x_{u_1} \big ] \bigg | X^{x,(1)}_s \bigg ]\\= & {} E\bigg [g_1\big ( X^{x,(1)}_{u_1} \big ) E\big [ g_2\big ( X^{x,(1)}_{u_2} \big ) \big | X^{x,(1)}_{u_1} \big ] \bigg | X^{x,(1)}_s \bigg ]\\= & {} E\bigg [g_1\big ( X^{x,(1)}_{u_1} \big ) E\big [ g_2\big ( X^{y_1,(1)}_{u_2-u_1} \big ) \big ] \big |_{y_1 = X^{x,(1)}_{u_1}} \bigg | X^{x,(1)}_s \bigg ]\\= & {} E\bigg [ g_1\big ( X^{y_2,(1)}_{u_1-s} \big ) E\big [ g_2\big ( X^{y_1,(1)}_{u_2-u_1} \big ) \big ] \big |_{y_1 = X^{y_2,(1)}_{u_1-s}} \bigg ] \bigg |_{y_2 = X^{x,(1)}_s}\\= & {} E\bigg [g_1\big ( X^{y_2,(1)}_{u_1-s} \big ) E\big [ g_2\big ( X^{y_2,(1)}_{u_2-s} \big ) \big | X^{y_2,(1)}_{u_1-s} \big ] \bigg ] \bigg |_{y_2 = X^{x,(1)}_s}\\= & {} E\bigg [g_1\big ( X^{y,(1)}_{u_1-s} \big ) g_2\big ( X^{y,(1)}_{u_2-s} \big ) \bigg ] \bigg |_{y = X^{x,(1)}_s} \quad {\textit{a.s.}} \end{aligned}$$

Inductively, we also get for any $k \in {\mathbb {N}}$, $s \le u_1 \le \dots \le u_k \le t$ and bounded functions $g_1, \dots , g_k$,

$$\begin{aligned} E\bigg [ g_1\big ( X^{x,(1)}_{u_1} \big ) \cdots g_k\big ( X^{x,(1)}_{u_k} \big ) \bigg | X^{x,(1)}_s \bigg ] =E\bigg [g_1\big ( X^{y,(1)}_{u_1-s} \big ) \cdots g_k\big ( X^{y,(1)}_{u_k-s} \big ) \bigg ] \bigg |_{y = X^{x,(1)}_s} \quad {\textit{a.s.}} \end{aligned}$$

By considering cylinder sets in $\sigma \big ( X^{x,(1)}_u; u \in [s,t] \big )$, the monotone class theorem gives the conclusion. $\square$

Proof

(Proof of Proposition 3.12) Let $h_1$ be a bounded function defined on the path space of $X^{x,(1)}_u, u \in [s,t]$. Moreover, let $h_2$ and $h_3$ be bounded functions on ${\mathbb {R}}$. Lemma 6.7 leads

$$\begin{aligned}&E\bigg [ h_1 \big ( X^{x,(1)}_u; u \in [s,t] \big ) h_2 \big (X^{x,(2)}_s \big ) h_3 \big ( X^{x,(3)}_s \big ) \bigg | X^x_s \bigg ]\\&\quad = h_2 \big ( X^{x,(2)}_s \big ) h_3 \big ( X^{x,(3)}_s \big ) E \bigg [h_1 \big ( X^{x,(1)}_u; u \in [s,t] \big ) \bigg | X^x_s \bigg ]\\&\quad = h_2 \big ( X^{x,(2)}_s \big ) h_3 \big ( X^{x,(3)}_s \big ) E \bigg [h_1 \big ( X^{x,(1)}_u; u \in [s,t] \big ) \bigg | {\mathscr {F}}^x_s \bigg | X^x_s \bigg ]\\&\quad = h_2 \big ( X^{x,(2)}_s \big ) h_3 \big ( X^{x,(3)}_s \big ) E \bigg [ h_1 \big ( X^{y,(1)}_{u_k-s} ; u \in [s,t] \big ) \bigg ] \bigg |_{y = X^{x,(1)}_s}\\&\quad = E\bigg [ h_1 \big ( X^{y_1,(1)}_{u_k-s} ; u \in [s,t] \big ) h_2 (y_2 ) h_3 ( y_3 ) \bigg ] \bigg |_{(y_1, y_2, y_3) = X^x_s} \quad {\textit{a.s.}} \end{aligned}$$

Therefore, the monotone class theorem yields that

$$\begin{aligned}&E\bigg [ h( X^{x,(1)}_u, X^{x,(2)}_s, X^{x,(3)}_s; u \in [s,t] \big ) \bigg | X^x_s \bigg ] \\&\quad = E\bigg [ h \big ( X^{y_1,(1)}_{u-s}, y_2, y_3; u \in [s,t] \big ) \bigg ] \bigg |_{(y_1, y_2, y_3) = X^x_s} \quad {\textit{a.s.}} \end{aligned}$$

for any bounded function h. Note that $X^{x,(1)}_u(\omega ), u \in [s,t]$ completely determines the jumps of $N^{x_1}_u(\omega ), u \in [s,t)$. Thus, for any bounded measurable function f, we can conclude

$$\begin{aligned}&E\big [ f \big ( X^x_t \big ) \big | X^x_s \big ]\\&\quad = E\left[ \left. f\left( \begin{array}{ccc} X^{x,(1)}_s\\ \left( X^{x,(1)}_s(t-s) + X^{x,(2)}_s \right) e^{-\beta (t-s)} +\int _{[s,t)} \alpha (t-u)e^{-\beta (t-u)}dN_u^{x_1} \\ \left( X^{x,(1)}_s(t-s)^2 + 2X^{x,(2)}_s(t-s) + X^{x,(3)}_s\right) e^{-\beta (t-s)} + \int _{[s,t)} \alpha (t-u)^2e^{-\beta (t-u)}dN_u^{x_1} \\ \end{array}\right) \right| X^x_s \right] \\&\quad = E\left[ \left. f\left( \begin{array}{c} y_1\\ \left( y_1(t-s) + y_2 \right) e^{-\beta (t-s)} + \int _{[s,t)} \alpha (t-u)e^{-\beta (t-u)}dN_{u-s}^{y_1} \\ \left( y_1(t-s)^2 + 2y_2(t-s) + y_3\right) e^{-\beta (t-s)} +\int _{[s,t)} \alpha (t-u)^2e^{-\beta (t-u)}dN_{u-s}^{y_1} \\ \end{array}\right) \right] \right| _{(y_1, y_2, y_3) = X^x_s} \\&\quad = E\left[ \left. f\left( \begin{array}{c} y_1\\ \left( y_1(t-s) + y_2 \right) e^{-\beta (t-s)} + \int _{[0,t-s)} \alpha (t-s-u)e^{-\beta (t-s-u)}dN_u^{y_1} \\ \left( y_1(t-s)^2 + 2y_2(t-s) + y_3\right) e^{-\beta (t-s)} +\int _{[0,t-s)} \alpha (t-s-u)^2e^{-\beta (t-s-u)}dN_u^{y_1} \\ \end{array}\right) \right] \right| _{(y_1, y_2, y_3) = X^x_s} \\&\quad = E\big [ f \big ( X^y_{t-s} \big ) \big ] \big |_{y = X^x_s} = \int _{{\mathbb {R}}^3} f(y) dP^{t-s}(X^x_s, dy)\quad {\textit{a.s.}} \end{aligned}$$

$\square$

To prove Proposition 3.13, we prepare the following lemma.

Lemma 6.8

For any $0 \le s < t$ and a bounded measurable function f,

$$\begin{aligned} E\left[ f(X^x_{t-s})\right] |_{x=\bar{X}_s(\bar{\omega })} =\bar{E}\left[ f(\bar{X}_t)|\bar{X}_s\right] (\bar{\omega }) \quad {\textit{a.s.}}\quad \bar{\omega } \end{aligned}$$

Proof

We again set $g(x, t) = e^{M_1x}e^{K_1t}$ and the operator $\bar{{\mathscr {A}}}$ same as Proposition 3.5. Denote the i-th jump time of $\bar{N}_t$ from time zero by $\tau _i$, i.e. $\tau _i = \inf \{t \ge 0 \ | \ \bar{N}[ (0,t] ] =i\}$. Then, $\int _{(0, t]} g(\bar{\lambda }^{(1)}_s + \alpha , s) -g(\bar{\lambda }^{(1)}_s, s) \big (d\bar{N}_s - \bar{\lambda }^{(1)}_s ds \big )$ is a $\tau _i$-local martingale, see Theorem 18.7 in Liptser and Shiryaev (2000). In the same way as the proof of Proposition 3.5, we get,

$$\begin{aligned} g\big (\bar{\lambda }^{(1)}_{t \wedge \tau _i}, t \wedge \tau _i \big ) -g\big ( \bar{\lambda }^{(1)}_0, 0 \big )&= \int _{(0, t \wedge \tau _i]} g(\bar{\lambda }^{(1)}_s + \alpha , s) -g(\bar{\lambda }^{(1)}_s, s) \big (d\bar{N}_s - \bar{\lambda }^{(1)}_s ds \big )\\&\quad +\int _{(0, t \wedge \tau _i]} \bar{{\mathscr {A}}}g(\bar{\lambda }^{(1)}_s, s)ds \quad {\textit{a.s.}} \end{aligned}$$

Thus, we have

$$\begin{aligned}&\bar{E}\left[ g(\bar{\lambda }^{(1)}_{t\wedge \tau _i}, t \wedge \tau _i) -g(\bar{\lambda }^{(1)}_0, 0) \Big | \bar{\lambda }^{(1)}_0 = x \right] \\&\quad = E\left[ \int _{(0, t \wedge \tau _i]} \bar{{\mathscr {A}}} g(\bar{\lambda }_s^{(1)}, s) ds \bigg | \bar{\lambda }^{(1)}_0 = x \right] \le \frac{K_2}{K_1}\left( e^{K_1t} - 1\right) . \end{aligned}$$

From the Fatou’s lemma, we have

$$\begin{aligned} \bar{E}\left[ e^{M_1\bar{\lambda }^{(1)}_t}e^{K_1t} \Big | \bar{\lambda }^{(1)}_0 = x \right] - e^{M_1x} \le \frac{K_2}{K_1}\left( e^{K_1t} - 1\right) . \end{aligned}$$

Then, from the stationarity of $\bar{\lambda }^{(1)}_0$, we also get the finiteness of moments of $\bar{\lambda }^{(1)}_t$ by

$$\begin{aligned} \bar{E}\left[ e^{M_1\bar{\lambda }^{(1)}_t} \right] \le \frac{K_2}{K_1}. \end{aligned}$$

For the operator ${\mathscr {A}}$ same as (3.2), $\bar{E}\big [ p\big ( \bar{\lambda }^{(1)}_t \big ) \big | \bar{\lambda }^{(1)}_s \big ] =e^{(t-s){\mathscr {A}}}p \big (\bar{\lambda }^{(1)}_s\big )$ a.s. holds for any $p \in {\mathscr {P}}$ in the same way of the proofs for Lemmas 3.7, 3.8 and (3.3). These properties lead the Markovian property of $\bar{\lambda }^{(1)}_t$ as in the proof of Theorem 3.9. Furthermore, for almost every $\bar{\omega }$,

$$\begin{aligned} \bar{E}\left[ p(\bar{X}^{(1)}_t) \Big | \bar{X}^{(1)}_s \right] (\bar{\omega })= & {} e^{(t-s){\mathscr {A}}}p\left( \bar{\lambda }^{(1)}_s(\bar{\omega }) - \mu \right) \\= & {} e^{(t-s){\mathscr {A}}}p\left( \lambda ^{\bar{X}^{(1)}_s(\bar{\omega })}_0 - \mu \right) = E\left[ p(X^{x,(1)}_{t-s})\right] \Big |_{x=\bar{X}^{(1)}_s(\bar{\omega })}. \end{aligned}$$

Thus, similarly as the proofs of Lemma 6.7 and Proposition 3.12, we get the conclusion. $\square$

Proof

(Proof of Proposition 3.13) Let $t\ge 0$ and $A\in {\mathscr {B}}({\mathbb {R}}^3)$. By taking $s=0$, $f(x) = 1_A(x)$ and integrating both sides of the equation of Lemma 6.8;

$$\begin{aligned} \int _{\bar{\varOmega }\times \varOmega }1_A\left( X^x_t(\omega )|_{x=\bar{X}_0 (\bar{\omega })}\right) dP(\omega )d\bar{P}(\bar{\omega }) = \bar{P} \left[ \bar{X}_t \in A\right] = P^{\bar{X}}[A], \end{aligned}$$

where we used the stationarity of $\bar{X}$. The above left hand side equals

$$\begin{aligned} \int _{\bar{\varOmega }\times \varOmega }1_A\left( X^x_t(\omega )|_{x=\bar{X}_0 (\bar{\omega })}\right) dP(\omega )d\bar{P}(\bar{\omega })= & {} \int _{{\mathbb {R}}^3_+}\int _{\varOmega }1_A\left( X^x_t(\omega )\right) dP(\omega )d P^{\bar{X}}(x) \\= & {} \int _{{\mathbb {R}}^3_+}P^t(x,A)dP^{\bar{X}}(x), \end{aligned}$$

and then we are done. $\square$

1.4.2 Ergodicity

The V-geometric ergodicity has been proved for the process $X^{(1)}$, see Proposition 4.5 in Clinet and Yoshida (2017). For the Hawkes core process $X = (X^{(1)}, X^{(2)}, X^{(3)})$, we can also prove it in a similar way. That is, we apply Theorem 6.1 in Meyn and Tweedie (1993). First, we again consider the extended generator and the drift criterion. The following lemma is proved by the same method as Proof of Proposition 4.5. in Clinet and Yoshida (2017).

Lemma 6.9

Let $\alpha , \beta$ and $\mu$ be the parameters of the Hawkes process $N^{x_1}_t$. For a differentiable function $f : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$, we define the operator ${\mathscr {A}}_X$ by

$$\begin{aligned} {\mathscr {A}}_Xf(y)&= (\mu + y_1) \left\{ f \left( y + \left( \begin{array}{c} \alpha \\ 0\\ 0 \end{array}\right) \right) - f ( y )\right\} \\&\quad + \big ( \partial _yf(y) \big )' \left\{ - \beta y + \left( \begin{array}{c} 0\\ y_1\\ 2y_2 \end{array}\right) \right\} ,\quad y = \left( \begin{array}{c} y_1\\ y_2\\ y_3 \end{array}\right) \in {\mathbb {R}}^3. \end{aligned}$$

Then, there exist a positive constant vector $M = (M_1, M_2, M_3)$ and positive constants $K_1, K_2$ such that for $V(y) = e^{My}$,

$$\begin{aligned} {\mathscr {A}}_XV(y) \le -K_1V(y) + K_2. \end{aligned}$$

Then, we can prove Proposition 3.14 with the help of this operator ${\mathscr {A}}_X$.

Proof

(Proof of Proposition 3.14) Now, it is proved in the completely same way as the proof of Proposition 3.5 replaced $g(x, t) = e^{M_1x}e^{K_1t}$ and $\bar{{\mathscr {A}}}$ by $g_X(x, t) = e^{Mx}e^{K_1t}$ and $\bar{{\mathscr {A}}}_X$ satisfying $\bar{{\mathscr {A}}}_Xg_X(x, t) =e^{K_1t}({\mathscr {A}}_Xe^{Mx} + K_1e^{Mx})$ respectively. $\square$

Second, we need to show that every compact set is petite for some skeleton chain, i.e. there exists $\delta >0$ such that for any compact set $C \in {\mathscr {B}}({\mathbb {R}}^3)$, we can choose a probability measure a on ${\mathbb {Z}}_+$ and a non-trivial measure $\phi _a$ on ${\mathbb {R}}^3$ such that

$$\begin{aligned} \sum _{n \in {\mathbb {Z}}_+} P^{\delta n}(x, A) a[n] \ge \phi _a[A] \quad \text {for all}\quad x \in C\quad \text {and}\quad A \in {\mathscr {B}}({\mathbb {R}}^3). \end{aligned}$$

The following concepts are closely related to petite sets. We call $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is an irreducible, if there exists a finite measure $\phi$ on ${\mathscr {B}}({\mathbb {R}}^3)$ such that if $\phi [A] > 0$ then

$$\begin{aligned} \sum _{n = 1}^{\infty } P^{\delta n}(x, A) > 0 \quad \text {for any}\quad x \in {\mathbb {R}}^3_+. \end{aligned}$$

Moreover, we call $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is a T-chain, if there exist $k \in {\mathbb {Z}}_+$ and non-trivial kernel T such that

• $T(x, {\mathbb {R}}^3) > 0$ for any $x \in {\mathbb {R}}^3_+$,

• $x \mapsto T(x, A)$ is lower semi-continuous for any $A \in {\mathscr {B}}({\mathbb {R}}^3)$,

• $P^{\delta k}(x, A) \ge T(x, A)$ for any $x \in {\mathbb {R}}^3_+$ and $A \in {\mathscr {B}}({\mathbb {R}}^3)$.

We consider a relation between the existence of petite compact sets and T-chain properties. The following lemma is well known, see Theorem 3.2 in Meyn and Tweedie (1992).

Lemma 6.10

Suppose that $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is an irreducible T-chain. Then, every compact set is petite.

Furthermore, we call $x^* \in {\mathbb {R}}^3$ is reachable, if for any open set $G \in {\mathscr {B}}({\mathbb {R}}^3)$ with $x^* \in G$,

$$\begin{aligned} \sum _{n=0}^{\infty }P^{\delta n}(y, G) > 0 \quad \text {for any}\quad y \in {\mathbb {R}}^3_+. \end{aligned}$$

Proof

(Proof of Proposition 3.15) We only have to prove that there exists $\delta > 0$ such that $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is an irreducible T-chain. First, we check the T-chain property.

Denote the i-th jump time of $N_t^x$ by $\tau ^x_i$. Let $\varDelta \tau _i^x$ be the interval time between the $(i-1)$-th and i-th jump of $N^x_t$, i.e. $\varDelta \tau _i^x = \tau ^x_i -\tau ^x_{i-1}$. As mentioned in Lemma A.4 of Clinet and Yoshida (2017), $\varDelta \tau _i^x$ has the conditional probability density (with respect to Lebesgue measure)

$$\begin{aligned} f^{\varDelta \tau _i^x} \big ( t \big | X^x_{\tau ^x_{i-1}} = y \big ) =\left( \mu + y_1e^{-\beta t} \right) \exp \left( \int _0^t \mu + y_1e^{-\beta s} ds \right) , \end{aligned}$$

where $y = (y_1, y_2, y_3)' \in {\mathbb {R}}^3_+$. Moreover, it is known that

$$\begin{aligned} f^{(\varDelta \tau _1^x, \dots , \varDelta \tau _i^x)} ( t_1, \dots , t_i | y)&= f^{\varDelta \tau _i^x} \big ( t_i | X^x_{\tau ^x_{i-1}} = X( t_1, \dots , t_{i-1} | y) \big ) \\&\quad \times f^{\varDelta \tau _{i-1}^x} \big ( t_{i-1} | X^x_{\tau ^x_{i-2}} = X( t_1, \dots , t_{i-2} | y) \big ) \\&\quad \times \cdots \times f^{\varDelta \tau _1^x} \big ( t_1 | X^x_{\tau ^x_1} = y \big ), \end{aligned}$$

where denote $\sum _{k=i}^j t_k$ by $T_{(i, j)}$ and

$$\begin{aligned} X( t_1, \dots , t_j | y)= & {} \left( \begin{array}{c} X^{(1)} ( t_1, \dots , t_j | y)\\ X^{(2)} ( t_1, \dots , t_j | y)\\ X^{(3)} ( t_1, \dots , t_j | y) \end{array}\right) \\= & {} \left( \begin{array}{c} y_1e^{-\beta T_{(1, j)}} + \sum _{l=1}^j \alpha e^{-\beta T_{(l+1, j)}} \\ \left( y_1T_{(1, j)} + y_2 \right) e^{-\beta T_{(1, j)}} + \sum _{l=1}^j \alpha T_{(l+1, j)} e^{ -\beta T_{(l+1, j)} } \\ \left( y_1T_{(1, j)}^2 + 2y_2T_{(1, j)} + y_3 \right) e^{-\beta T_{(1, j)}} + \sum _{l=1}^j \alpha T_{(l+1, j)}^2 e^{ -\beta T_{(l+1, j)} } \end{array}\right) . \end{aligned}$$

Note that $f^{(\varDelta \tau _1^x, \dots , \varDelta \tau _i^x)} ( t_1, \dots , t_i | y)$ is obviously smooth in y. Then, for any $\delta > 0$ and $A \in {\mathscr {B}}({\mathbb {R}}^3)$,

$$\begin{aligned} P^{\delta }(x, A)= & {} P\left[ X^x_{\delta } \in A \right] \\\ge & {} P\left[ X^x_{\delta } \in A , \ \sharp \{j | \tau ^x_j< \delta \} = 3 \right] \\= & {} \int _{{\mathbb {R}}_+^4} 1_{\{ \check{X}(\delta ; t_1, t_2, t_3 | x) \in A \}} 1_{\{ T_{(1,3)} < \delta \} \cap \{ T_{(1,4)} \ge \delta \}} f^{(\varDelta \tau _1^x, \varDelta \tau _2^x, \varDelta \tau _3^x, \varDelta \tau _4^x)} (t_1, t_2, t_3, t_4 | x) dt_1dt_2dt_3dt_4, \end{aligned}$$

where

$$\begin{aligned} \check{X}(\delta ; t_1, t_2, t_3 | x) = \left( \begin{array}{c} x_1e^{-\beta \delta } + \sum _{l=1}^3 \alpha e^{-\beta (\delta - T_{(1, l)})} \\ \left( x_1\delta + x_2 \right) e^{-\beta \delta } + \sum _{l=1}^3 \alpha (\delta - T_{(1, l)}) e^{ -\beta (\delta - T_{(1, l)}) } \\ \left( x_1\delta ^2 + 2x_2\delta + x_3 \right) e^{-\beta \delta } +\sum _{l=1}^3 \alpha (\delta - T_{(1, l)})^2 e^{ -\beta (\delta - T_{(1, l)}) } \end{array}\right) \end{aligned}$$

and it is obviously smooth in x. However, the indicator function $1_{\{ \check{X}(\delta ; t_1, t_2, t_3 | x) \in A \}}$ is not always lower semi-continuous in x. Thus, we consider a change of variable for the map $H_{x,\delta } : (t_1, t_2, t_3) \mapsto \check{X}(\delta ; t_1, t_2, t_3 | x)$, as in Proof of Lemma A.3 of (Clinet and Yoshida 2017). Denote the Jacobian matrix of $H_{x,\delta }$ at $(t_1, t_2, t_3)$ by $J_{\delta }(t_1, t_2, t_3)$. Then, completely elementary calculations leads

$$\begin{aligned} J_{\delta }(t_1, t_2, t_3) = (J_{i,j})_{i,j = 1,2,3}, \end{aligned}$$

where for $j=1,2,3$

$$\begin{aligned} J_{1,j} = \sum _{l=j}^3 \alpha \beta e^{-\beta (\delta - T_{(1,l)})}, \ J_{2,j} = \sum _{l=j}^3 \alpha \left\{ \beta (\delta - T_{(1,l)}) - 1 \right\} e^{-\beta (\delta - T_{(1,l)})}, \\ \text {and} \quad J_{3,j} = \sum _{l=j}^3 \alpha \left\{ \beta (\delta -T_{(1,l)})^2 - 2 (\delta - T_{(1,l)}) \right\} e^{-\beta (\delta - T_{(1,l)})}. \end{aligned}$$

The determinant of the Jacobian matrix has the following representation.

$$\begin{aligned} |J_{\delta }(t_1, t_2, t_3)|= & {} \prod _{l=1}^3 \alpha \beta e^{-\beta (\delta - T_{(1,l)})} \times \left| \begin{array}{ccc} 1 &{} 1&{} 1\\ \big ( \delta - T_{(1,1)} \big )&{} \big ( \delta - T_{(1,2)} \big ) &{} \big ( \delta - T_{(1,3)} \big ) \\ \big ( \delta - T_{(1,1)} \big )^2 &{} \big ( \delta - T_{(1,2)} \big )^2 &{} \big ( \delta - T_{(1,3)} \big )^2 \\ \end{array}\right| . \end{aligned}$$

It is a Vandermonde determinant and thus not zero if $(t_1, t_2, t_3) = (\tau , \tau , \tau )$ for $\tau \in (0,\delta /3)$. We consider a neighborhood at the such point $(t_1, t_2, t_3) = (\tau , \tau , \tau )$. Set $B(t_1, t_2, t_3, t_4) = \{ T_{(1,3)} < \delta \} \cap \{ T_{(1,4)} > \delta \} \cap \{(t_1, t_2 ,t_3) \in (\tau -\varepsilon ,\tau +\varepsilon )^3\}$ for sufficient small $\varepsilon >0$. Then, we get a non-trivial component T(x, A) as below.

$$\begin{aligned} P^{\delta }(x, A)&\ge \int _{{\mathbb {R}}_+^4} 1_{\{ \check{X}(\delta ; t_1, t_2, t_3 | x) \in A \}} 1_{\{ T_{(1,3)} < \delta \} \cap \{ T_{(1,4)} \ge \delta \}}\\&\quad f^{(\varDelta \tau _1^x, \varDelta \tau _2^x, \varDelta \tau _3^x, \varDelta \tau _4^x)} (t_1, t_2, t_3, t_4 | x) dt_1dt_2dt_3dt_4\\&\ge \int _{{\mathbb {R}}_+^4} 1_{\{ (y_1, y_2, y_3) \in A \}} 1_{B (H^{-1}_{x, \delta }(y_1, y_2, y_3), t_4)} f^{(\varDelta \tau _1^x, \varDelta \tau _2^x, \varDelta \tau _3^x, \varDelta \tau _4^x)} (H^{-1}_{x, \delta }(y_1, y_2, y_3), t_4 | x) \\&\quad \left| J_{\delta } \left( H^{-1}_{x, \delta }(y_1, y_2, y_3) \right) \right| ^{-1} dy_1dy_2dy_3dt_4 =: T(x, A). \end{aligned}$$

Since $B(t_1, t_2, t_3, t_4)$ is a countable union of open intervals, continuity of $H^{-1}_{x,\delta }$ in x leads that $x \mapsto T(x,A)$ is lower semi-continuous. Thus, $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is a T-chain.

Finally, we prove that $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is irreducible. Since $\{X^x_{\delta n}\}_{n \in {\mathbb {Z}}_+}$ is a T-chain, we only have show that there exists a reachable point $x^* \in {\mathbb {R}}^3_+$, i.e. for any open set $O \in {\mathscr {B}}({\mathbb {R}}^3)$ containing $x^*$,

$$\begin{aligned} \sum _{n=0}^{\infty } P^{\delta n}(y, O) > 0 \quad \text {for any}\quad y \in {\mathbb {R}}^3_+, \end{aligned}$$

see Proposition 6.2.1 in Meyn and Tweedie (1993). However, we can easily show that (0, 0, 0) is a reachable point. Indeed, if a jump will never occur, for any neighborhood O of (0, 0, 0), $X^x_{\delta n} \in O$ for sufficient large $n \in {\mathbb {N}}$. By the form of $f^{\varDelta \tau _1^x} \big ( t \big | x\big )$, the probability there is no jump on $[0, \delta n]$ is positive. Thus, we get the conclusion. $\square$

1.5 Proofs of Section 4

In this subsection, we will prove Theorem 4.6. For this purpose, it is enough to confirm that there exist some constants satisfying (2.15) and the conditions [A1]–[A3], [B0]–[B4], [C1] hold. We explain each condition separately by dividing each small section.

1.5.1 Proof of Proposition 4.4 (condition [A1])

In Markovian framework, as mentioned in Kusuoka and Yoshida (2000) and Yoshida (2004), the mixing property is derived from the ergodicity. Concretely, the geometric mixing property is reduced to the following property;

[A1$'$:

] There exists a positive constant a such that

$$\begin{aligned} \sup _{\begin{array}{c} f \in {\mathscr {F}}{\mathscr {B}}_{[t, \infty )} \\ : \left\| f \right\| _{\infty } \le 1 \end{array}} \left\| E\left[ f \left| X_s\right. \right] - E[f] \right\| _{L^1(P)} < a^{-1}e^{-a(t-s)} \quad \text {for any}\quad t> s > 0. \end{aligned}$$

Proposition 6.11

The Markovian property in Proposition 3.11 and [A1$'$] lead [A1].

Proof

For any $f \in {\mathscr {F}}{\mathscr {B}}_{[0,s]}$ and $g \in {\mathscr {F}}{\mathscr {B}}_{[t,\infty )}$ with $\left\| f \right\| _{\infty } \le 1$ and $\left\| g \right\| _{\infty } \le 1$,

$$\begin{aligned} \left| E[fg] - E[f]E[g] \right|= & {} \left| E\left[ f(g-E[g])\right] \right| = \left| E\left[ fE\left[ g-E[g]\left| {\mathscr {B}}_{[0,s]}\right. \right] \right] \right| \\\le & {} \left\| E\left[ g-E[g]\left| {\mathscr {B}}_{[0,s]}\right. \right] \right\| _{L^1(P)} = \left\| E\left[ g\left| X_s\right. \right] - E[g]\right\| _{L^1(P)} \\\le & {} a^{-1}e^{-a(t-s)}. \end{aligned}$$

$\square$

Proof

(Proof of Proposition 4.4) We confirm that [A1$'$] follows from Proposition 3.15. Let $s \le t$ and $f \in {\mathscr {F}}{\mathscr {B}}_{[t,\infty )}$ with $\Vert f\Vert _{\infty } \le 1$. From the Markovian property, we have

$$\begin{aligned} E\left[ f \left| X_s \right. \right] = E\left[ E\left[ f \left| {\mathscr {B}}_{[0,t]}\right. \right] \left| X_s \right. \right] = E\left[ E\left[ f \left| X_t \right. \right] \left| X_s \right. \right] . \end{aligned}$$

There exists a measurable function g such that $E\left[ f \left| X_t \right. \right] = g(X_t)$ and $\Vert g\Vert _{\infty } \le 1$. From Proposition 3.12, we get

$$\begin{aligned} E\left[ f \left| X_s \right. \right] = E\left[ g(X_t) \left| X_s \right. \right] = \int _{{\mathbb {R}}^3} g(y) P^{t-s}(X_s, dy). \end{aligned}$$

On the other hand, we have

$$\begin{aligned} E\left[ f\right] = E\left[ g(X_t)\right] = \int _{{\mathbb {R}}^3} g(y) P^t(X_0, dy). \end{aligned}$$

Therefore, by using Proposition 3.15,

$$\begin{aligned}&\sup _{\begin{array}{c} f \in {\mathscr {F}}{\mathscr {B}}_{[t, \infty )} \\ : \left\| f \right\| _{\infty } \le 1 \end{array}} \left\| E\left[ f \left| X_s\right. \right] - E[f] \right\| _{L^1(P)} \\&\quad \le \sup _{g : \Vert g \Vert _{\infty } \le 1} \left\| \int _{{\mathbb {R}}^3} g(y) P^{t-s}(X_s, dy) - \int _{{\mathbb {R}}_+} g(y) P^{t}(X_0, dy)\right\| _{L^1(P)} \\&\quad \le \sup _{g : \Vert g \Vert _{\infty } \le 1} \left\{ \left\| \int _{{\mathbb {R}}^3} g(y) \left( P^{t-s}(X_s, dy) - P^{\bar{X}}(dy) \right) \right\| _{L^1(P)}\right. \\&\qquad \left. + \left\| \int _{{\mathbb {R}}^3} g(y) \left( P^{t}(X_0, dy) - P^{\bar{X}}(dy) \right) \right\| _{L^1(P)} \right\} \\&\quad \le E\left[ \left\| P^{t-s}(X_s, \cdot ) - P^{\bar{X}} \right\| _{e^{M\cdot }}\right] + E\left[ \left\| P^{t}(X_0, \cdot ) -P^{\bar{X}}\right\| _{e^{M\cdot }}\right] \\&\quad \le E\left[ B(e^{M X_s } + 1)r^{t-s} \right] +E\left[ B(e^{M X_0} + 1)r^{t}\right] \\&\quad = r^{t-s}B\left( E\left[ e^{M X_s} \right] + 1 + 2r^s\right) . \end{aligned}$$

Finally, from Proposition 3.14, we may choose sufficient small $a>0$ that satisfies [A1$'$]. $\square$

1.5.2 Condition [A2]

$Z_0 \in \bigcap _{p>1}L^p(P)$ and $P[Z_0]=0$ are obvious. We can write each component of $Z^t_{t+h}$ as

$$\begin{aligned} \int _t^{t+h} \frac{p_1(X_s)}{\lambda _s^2} d\tilde{N}_s +\int _t^{t+h} \frac{p_2(X_s)}{\lambda _s^2} ds - E \left[ \int _t^{t+h} \frac{p_2(X_s)}{\lambda _s^2} ds \right] \end{aligned}$$

where $p_1$ and $p_2$ are 3-variable polynomial functions. From Proposition 3.14, we have $\sup _t ||X_t||_{L^p(P)} < \infty$ for any $p>1$. By considering $t \in [0, T]$ for an arbitrary $T>0$, $\int _0^t p_1(X_s)/\lambda _s^2 d\tilde{N}_s$ is a square integrable martingale, see Theorem 18.8 in Liptser and Shiryaev (2000). Thus, we immediately get $E\big [Z^t_{t+\varDelta }\big ] = 0$ for any $\varDelta >0$ and $t > 0$.

The rest of the proof is $\sup _{t \in {\mathbb {R}}_+, 0 \le h \le \varDelta } \left\| Z^t_{t+h}\right\| _{L^p(P)} < \infty$. When we consider the $L^p$ boundedness, it is enough to consider the form of $p = 2^k$ for $k \in {\mathbb {N}}$. We get

$$\begin{aligned} \left\| E\left[ \int _t^{t+h} \frac{p_2(X_s)}{\lambda _s^2} ds \right] \right\| _{L^p(P)} < \frac{\sup _s E[|p_2(X_s)|] }{\mu _0^2} h. \end{aligned}$$

Moreover, since $h^{-1} ds$ is a probability measure on $[t, t+h]$, by the Jensen’s inequality,

$$\begin{aligned} \left\| \int _t^{t+h} \frac{p_2(X_s)}{\lambda _s^2} ds \right\| _{L^p(P)}\le & {} \left( E\left[ \int _t^{t+h} \left( \frac{p_2(X_s)}{\lambda _s^2}h \right) ^p h^{-1}ds \right] \right) ^{\frac{1}{p}}\\\le & {} \frac{\sup _s \Vert p_2(X_s)\Vert _{L^p(P)}}{\mu _0^2} h. \end{aligned}$$

On the other hand,

$$\begin{aligned} {\mathscr {M}}_h = \int _t^{t+h} \frac{p_1(X_s)}{\lambda _s^2} d\tilde{N}_s \end{aligned}$$

is also a square integrable martingale. Then, the Burkholder–Davis–Gundy inequality leads that there exists a positive constant $C_k$ (take again new $C_k$ in the last step) such that

$$\begin{aligned} E\left[ \left| {\mathscr {M}}_h \right| ^{2^k}\right]&\le C_k E\left[ \left| [{\mathscr {M}} ]_h \right| ^{2^{k-1}}\right] \nonumber \\&= C_k E\left[ \left| \int ^{t+h}_t \left( \frac{p_1(X_s)}{\lambda _s^2} \right) ^2 dN_s \right| ^{2^{k-1}}\right] \nonumber \\&\le C_k \left( E\left[ \left| \int ^{t+h}_t \left( \frac{p_1(X_s)}{\lambda _s^2} \right) ^2 d\tilde{N}_s \right| ^{2^{k-1}}\right] \right. \\&\quad \left. + E\left[ \left| \int ^{t+h}_t \left( \frac{p_1(X_s)}{\lambda _s^2} \right) ^2 \lambda _sds \right| ^{2^{k-1}}\right] \right) , \end{aligned}$$

where $[{\mathscr {M}}]_h$ represents the quadratic variation of ${\mathscr {M}}_h$. We used the Jensen’s inequality in the last estimation. By induction, one gets some constant $Q_k$ (take again new $Q_k$ in the last step) such that

$$\begin{aligned} E\left[ \left| {\mathscr {M}}_h \right| ^{2^k}\right]\le & {} Q_k\sum _{j = 1}^k E\left[ \left| \int ^{t+h}_t \left( \frac{p_1(X_s)}{\lambda _s^2} \right) ^{2^j} \lambda _sds \right| ^{2^{k-j}}\right] \nonumber \\\le & {} Q_k\sum _{j = 1}^k E\left[ \int ^{t+h}_t \left( \frac{p_1(X_s)}{\lambda _s^2} \right) ^{2^k} \lambda _s^{2^{k-j}} h^{2^{k-j}} h^{-1}ds \right] \nonumber \\\le & {} Q_k (h + 1)^{2^{k-1}}. \end{aligned}$$

(6.17)

Therefore, for any $\varDelta > 0$ and $p>0$, $\sup _{t \in {\mathbb {R}}_+, 0 \le h \le \varDelta } \left\| Z^t_{t+h}\right\| _{L^p(P)} < \infty$ holds. Then, the condition [A2] is verified.

1.5.3 Condition [A3]

The condition [A3] follows from Lemma 3.15. and the proof of Lemma A.7. in Clinet and Yoshida (2017).

1.5.4 Condition [B0]

(i), (ii) and (iv) are obvious. (iii) immediately follows a square integrable martingale property:

$$\begin{aligned} {\textit{Cov}}\left[ \int _0^t \frac{p_1(X_s)}{\lambda _s} d\tilde{N}_s , \int _0^t \frac{p_2(X_s)}{\lambda _s} d\tilde{N}_s \right]= & {} E\left[ \int _0^t \frac{p_1(X_s)p_2(X_s)}{\lambda _s^2} d[\tilde{N}]_s \right] \\= & {} E\left[ \int _0^t \frac{p_1(X_s)p_2(X_s)}{\lambda _s} ds \right] \end{aligned}$$

for any 3-variable polynomial functions $p_1$ and $p_2$.

1.5.5 Condition [B1]

We take any constant $L>1$. From (4.1) and (6.17), we immediately deduce that for any $k \in {\mathbb {N}}$

$$\begin{aligned} E \left[ \left| T^{-\frac{1}{2}} l_a(\theta _0)\right| ^{2^k} \right] \le Q_k (1 + T^{-1})^{2^{k-1}}. \end{aligned}$$

Therefore, the condition [B1] holds for any $q_1 > 1$. In particular, we can choose $q_1$ satisfying $q_1 > 3L$.

1.5.6 Condition [B2]

Let $L, q_1$ and $q_3$ be positive constants with $L>1, q_1 > 3L$ and $q_3 > \frac{q_1L}{q_1-3L}$. We arbitrary set a positive constant $q_2$ with $q_2 > \max \left( 3, \frac{3q_1L}{q_1-3L}\right)$ for given constants L and $q_1$. Let $Y_t(\theta ) = \left( X_t^{(1)}(\theta _0), X_t^{(1)}(\theta ), X_t^{(2)}(\theta ), X_t^{(3)}(\theta ), X_t^{(4)}(\theta ) \right)$ for $\theta = (\mu , \alpha , \beta )$. From the relation

$$\begin{aligned} \partial _{\theta } X_t^n(\theta ) = \left( \begin{array}{c} 0 \\ \alpha ^{-1}X_t^n(\theta ) \\ -X_t^{n+1}(\theta ) \end{array}\right) \end{aligned}$$

and a verification of the permutation rule of the symbol $\partial _{\theta }$ and $\int _0^T$, we can write, for both of the case $k = 2$ and $k = 3$,

$$\begin{aligned}&T^{\frac{\gamma }{2}}\left( T^{-1}l_{a_1 \cdots a_k}(\theta ) -\nu _{a_1 \cdots a_k}(\theta ) \right) \\&\quad = T^{\frac{\gamma }{2} -1} \int _0^T \frac{p_1(Y_s(\theta ))}{\lambda _s^4(\theta )} d\tilde{N}_s +T^{\frac{\gamma }{2}} \left\{ \frac{1}{T}\int _0^T \frac{p_2(Y_s(\theta ))}{\lambda _s^4(\theta )} ds -E\left[ \frac{1}{T} \int _0^T \frac{p_2(Y_s(\theta ))}{\lambda _s^4(\theta )} ds \right] \right\} \end{aligned}$$

with some polynomial functions $p_1$ and $p_2$. Lemma A.5. in Clinet and Yoshida (2017) guarantees

$$\begin{aligned} \sup _t \sum _{i=0}^4 \Vert \sup _{\theta \in \varTheta } \partial _{\theta }^i \lambda _t(\theta ) \Vert _{L^p(P)} < \infty \end{aligned}$$

(6.18)

for any $p>1$. Thus, we have $\sup _t ||\sup _{\theta \in \varTheta } Y_t(\theta )||_{L^p(P)} < \infty$ for any $p>1$. Moreover, $Y_t(\theta )$ is $\sigma \left( N_s ; s \le t \right)$-predictable. From the restriction of (2.15), $\frac{\gamma }{2} -1 <-\frac{1}{2}$ holds. Then, in the same method of the proof of the condition [A2], we can see that

$$\begin{aligned} \sup _{T>0, \theta \in \varTheta } E\left[ \left| T^{\frac{\gamma }{2} -1} \int _0^T \frac{p_1(Y_s(\theta ))}{\lambda _s^4(\theta )} d\tilde{N}_s\right| ^{2^k} \right] < \infty \end{aligned}$$

for any $k \in {\mathbb {N}}$.

The later term is estimated by using the ergodicity of $X_t^{(1)}(\theta _0)$. Let

$$\begin{aligned} \tilde{Y}(s, t, \theta )= & {} \Bigg (X_t^{(1)}(\theta _0), \int _{(s,t)} \alpha e^{-\beta (t-u)}dN_u^{x_1}, \int _{(s,t)} \alpha (t-u) e^{-\beta (t-u)}dN_u^{x_1}, \\&\int _{(s,t)} \alpha (t-u)^2 e^{-\beta (t-u)}dN_u^{x_1}, \int _{(s,t)} \alpha (t-u)^3 e^{-\beta (t-u)}dN_u^{x_1} \Bigg ). \end{aligned}$$

Denote $D_{\uparrow }\big ({\mathbb {R}}_+^5, {\mathbb {R}}\big )$ as the set of functions $\psi : {\mathbb {R}}_+^5 \rightarrow {\mathbb {R}}$ that satisfy:

• $\psi$ are of class $C^1({\mathbb {R}}_+^5)$.

• $\psi$ and $|\bigtriangledown \psi |$ are polynomial growth.

By replacing $X^{\alpha }(t, \theta )$ by $Y_t(\theta )$ and $\tilde{X}^{\alpha }(s, t, \theta )$ by $\tilde{Y}(s, t, \theta )$ in the proof of Lemma A.6. and using Lemma 3.16. in Clinet and Yoshida (2017), we can get the following ergodicity property: There exist a mapping $\pi : D_{\uparrow }\big ({\mathbb {R}}_+^5, {\mathbb {R}}\big ) \times \varTheta \rightarrow {\mathbb {R}}$ and a constant $\gamma ' \in \left( 0, \frac{1}{2} \right)$ such that for any $\psi \in D_{\uparrow }\big ({\mathbb {R}}_+^5, {\mathbb {R}}\big )$ and for any $p>1$,

$$\begin{aligned} \sup _{\theta \in \varTheta } T^{\gamma '} \left\| \frac{1}{T} \int _0^T \psi (Y_s(\theta )) ds - \pi (\psi , \theta )\right\| _{L^p(P)} \rightarrow 0 \quad \text {as}\quad T \rightarrow \infty . \end{aligned}$$

However, in the case of the exponential Hawkes process, we can choose $\gamma ' \in \left( 0, \frac{1}{2} \right)$ arbitrarily. This arbitrariness follows from the fact $\Vert Y_t(\theta ) -\bar{Y}_t(\theta ) \Vert _{L_1(P)}$ is exponentially decreasing uniformly in $\theta$ for some stationary process $\bar{Y}_t(\theta )$, see the proof of the stability condition part in Lemma A.6. of Clinet and Yoshida (2017). Therefore, by taking $\gamma ' \in \left( 0, \frac{1}{2} \right)$ and $\gamma = 2\gamma '$ satisfying $\frac{2}{3} + \max \left( \frac{L}{q_2}, \frac{L}{3q_3}\right)< \gamma < 1 - \frac{L}{q_1}$, we get

$$\begin{aligned} \left\| T^{\frac{\gamma }{2}} \left\{ \frac{1}{T}\int _0^T \frac{p_2(Y_s(\theta ))}{\lambda _s^4(\theta )} ds - E \left[ \frac{1}{T} \int _0^T \frac{p_2(Y_s(\theta ))}{\lambda _s^4(\theta )} ds \right] \right\} \right\| _{L^p(P)} \rightarrow 0 \quad \text {as}\quad T \rightarrow \infty \end{aligned}$$

for any $p>1$. It means that the condition [B2] holds for any $q_2 >\max \left( 3, \frac{3q_1L}{q_1-3L}\right)$ and some $\gamma$ with $\frac{2}{3} + \max \left( \frac{L}{q_2}, \frac{L}{3q_3}\right)<\gamma < 1 - \frac{L}{q_1}$.

1.5.7 Condition [B3]

We only have to show that there exist an open set $\tilde{\varTheta }$ including $\theta _0$ and a positive constant $T_0$ such that

$$\begin{aligned} \inf _{T>T_0, \theta \in \tilde{\varTheta },|x|=1} \left| x' \nu _{a b} (\theta ) \right| > 0. \end{aligned}$$

(6.19)

Because, if (6.19) holds, continuity of $\nu _{a b}(\theta )$ and $x' \nu _{a b}(\theta ) \ne 0$ lead

$$\begin{aligned} \left| \int _0^1 x' \nu _{a b} (\theta _1 + s(\theta _2 - \theta _1))ds \right| > \inf _{\theta \in \tilde{\varTheta }} \left| x' \nu _{a b} (\theta ) \right| \end{aligned}$$

for any $\theta _1, \theta _2 \in \tilde{\varTheta }, T>T_0$ and x with $|x|=1$. Therefore, we consider to prove (6.19). We can write

$$\begin{aligned} \nu _{a b} (\theta ) = - E\left[ \frac{1}{T} \int ^T_0 \frac{\left( \partial _{\theta }\lambda _s (\theta ) \right) ^{\otimes 2}}{\lambda _s^2 (\theta )} \lambda _s(\theta _0) ds \right] + E\left[ \frac{1}{T} \int ^T_0 \frac{\partial _{\theta }^2 \lambda _s(\theta )}{\lambda _s(\theta )} \left( \lambda _s(\theta _0) - \lambda _s(\theta ) \right) ds \right] . \end{aligned}$$

With the help of (6.18), for the first term, we have

$$\begin{aligned}&\left| g_T - E\left[ \frac{1}{T} \int ^T_0 \frac{\left( \partial _{\theta }\lambda _s (\theta ) \right) ^{\otimes 2}}{\lambda _s^2(\theta )} \lambda _s(\theta _0) ds \right] \right| \\&\quad \le \frac{|\theta _0 - \theta | }{T} E\left[ \int ^T_0 \sup _{\theta \in \varTheta } \left| \partial _{\theta } \frac{\left( \partial _{\theta }\lambda _s (\theta ) \right) ^{\otimes 2}}{\lambda _s^2(\theta )} \right| \lambda _s(\theta _0) ds \right] \le C_{\varTheta , 1} |\theta _0 - \theta |, \end{aligned}$$

where $C_{\varTheta , 1}$ is a positive constant that does not depend on T. For the second term, we also get

$$\begin{aligned}&\left| E\left[ \frac{1}{T} \int ^T_0 \frac{\partial _{\theta }^2 \lambda _s(\theta )}{\lambda _s(\theta )} \left( \lambda _s(\theta _0) - \lambda _s(\theta ) \right) ds \right] \right| \\&\quad \le \frac{|\theta _0 - \theta | }{T} E\left[ \int ^T_0 \sup _{\theta \in \varTheta } \left| \frac{\partial _{\theta }^2 \lambda _s(\theta )}{\lambda _s^2(\theta )} \right| \sup _{\theta \in \varTheta } \left| \partial _{\theta }\lambda _s (\theta ) \right| ds \right] \le C_{\varTheta , 2} |\theta _0 - \theta |, \end{aligned}$$

where $C_{\varTheta , 2}$ is a positive constant independent of T. Since we have assumed $g_T$ is non-singular for large T in the condition [A3], we may choose $\tilde{\varTheta }$ and $T_0>0$ such that

$$\begin{aligned} \inf _{T>T_0, \theta \in \tilde{\varTheta },|x|=1} \left| x' \nu _{a b} (\theta ) \right|&\ge \inf _{T>T_0,|x|=1} \left| x' g_T \right| \\&\quad - \sup _{T>T_0, \theta \in \tilde{\varTheta }}\left| g_T - E\left[ \frac{1}{T} \int ^T_0 \frac{\left( \partial _{\theta }\lambda _s (\theta ) \right) ^{\otimes 2}}{\lambda _s^2(\theta )} \lambda _s(\theta _0) ds \right] \right| \\&\quad - \sup _{T>T_0, \theta \in \tilde{\varTheta }} \left| E\left[ \frac{1}{T} \int ^T_0 \frac{\partial _{\theta }^2 \lambda _s(\theta )}{\lambda _s(\theta )} \left( \lambda _s(\theta _0) - \lambda _s(\theta ) \right) ds \right] \right| > 0. \end{aligned}$$

1.5.8 Condition [B4]

Let L and $q_1$ be positive constants with $L>1$ and $q_1 > 3L$. We apply Sobolev’s inequality (see Theorem 4.12 of Adams and Fournier 2003). We take any integer $q_3 > \max \left( 3, \frac{q_1L}{q_1-3L}\right)$ and some constant $K(\varTheta , q_3)$ such that

$$\begin{aligned} E\left[ \sup _{\theta \in \varTheta } \left| \frac{1}{T} l_{a_1 \cdots a_4}(\theta ) \right| ^{q_3} \right]&\le K(\varTheta , q_3)\left\{ \int _{\varTheta } E\left[ \left| \frac{1}{T} l_{a_1 \cdots a_4}(\theta ) \right| ^{q_3} \right] d\theta \right. \\&\quad \left. + \int _{\varTheta } E\left[ \left| \frac{1}{T} \partial _{\theta } l_{a_1 \cdots a_4}(\theta ) \right| ^p \right] d\theta \right\} \\&\lesssim \sum _{k=1}^4 \sup _{\theta \in \varTheta } E\left[ \left| \frac{1}{T} l_{a_1 \cdots a_4}(\theta ) \right| ^{q_3} \right] . \end{aligned}$$

Let $Y'_t(\theta ) = \left( X_t^{(1)}(\theta _0), X_t^{(1)}(\theta ), X_t^{(2)}(\theta ), X_t^{(3)}(\theta ), X_t^{(4)}(\theta ), X_t^{(5)}(\theta ) \right)$. We may easily confirm that

$$\begin{aligned} \frac{1}{T}l_{a_1, \dots , a_4}(\theta ) = \frac{1}{T} \int _0^T \frac{p_1(Y'_s(\theta ))}{\lambda _s^{8}(\theta )} d\tilde{N}_s + \frac{1}{T} \int _0^T \frac{p_2(Y'_s(\theta ))}{\lambda _s^{8}(\theta )} ds \end{aligned}$$

with some polynomial functions $p_1$ and $p_2$. In a similar way as Lemma A.5 in Clinet and Yoshida (2017), we can prove that $\sup _t \Vert \sup _{\theta \in \varTheta }Y'_t(\theta ) \Vert _{L^p(P)} < \infty$ for any $p>1$. Then, like (6.17), we have

$$\begin{aligned} \sup _{T>0, \theta \in \varTheta } E\left[ \left| \frac{1}{T} \int _0^T \frac{p_1(Y'_s(\theta ))}{\lambda _s^{8}(\theta )} d\tilde{N}_s\right| ^{2^k} \right] < \infty \end{aligned}$$

for any $k \in {\mathbb {N}}$. On the other hand, by the Jensen’s inequality,

$$\begin{aligned} \sup _{T>0, \theta \in \varTheta } E\left[ \left| \frac{1}{T} \int _0^{T} \frac{p_2(Y'_s(\theta ))}{\lambda _s^{8}(\theta )} ds \right| ^p \right] \le \sup _{T>0} \frac{1}{T} \int _0^{T} E\left[ \sup _{\theta \in \varTheta } \left| \frac{p_2(Y'_s(\theta ))}{\lambda _s^{8}(\theta )} \right| ^p \right] ds < \infty \end{aligned}$$

for any $p>1$. Therefore, the condition [B4] holds for any constant $q_3 > \max \left( 3, \frac{q_1L}{q_1-3L}\right)$.

1.5.9 Condition [C1]

In Theorem 4.6 of Clinet and Yoshida (2017), the convergence of moments is proved for $\sqrt{T}\big (\hat{\theta }_T- \theta _0\big )$. The condition [C1] directly follows from this statement.