Appendix
In this appendix, we provide the proofs of Theorems 3.4 and 3.5 in Sect. 3. The following properties for the negative binomial distribution are useful for proving these two theorems. Based on Christou and Fokianos [7], for \(x\in \mathbb {N}_0\) and all \(\lambda >0\),
-
(P1)
\(0<P_{\theta }(x|\mathcal {F}_{t-1})<1\),
-
(P2)
\(\sum _{x=0}^{\infty }P_{\theta }(x|\mathcal {F}_{t-1})=1\),
-
(P3)
\(\sum _{x=0}^{\infty }xP_{\theta }(x|\mathcal {F}_{t-1})=\lambda \),
-
(P4)
\(\sum _{x=0}^{\infty }x^2P_{\theta }(x|\mathcal {F}_{t-1}) =(\sigma +1)\lambda ^2+\lambda \),
-
(P5)
\(\sum _{x=0}^{\infty }x^3P_{\theta }(x|\mathcal {F}_{t-1}) =(1+3\sigma +2\sigma ^2)\lambda ^3+3(1+\sigma )\lambda ^2+\lambda \),
where \(P_{\theta }(x|\mathcal {F}_{t-1})\) is defined by substituting \(X_t\) with x in \(P_{\theta }(X_t|\mathcal {F}_{t-1})\).
Lemma A.1
Under \((\mathrm{A}1){-}(\mathrm{A}3)\), we have
$$\begin{aligned} \sup _{\theta ^*\in \Theta ^*}|\lambda _t(\theta ^*)-{\tilde{\lambda }}_t(\theta ^*)|\le V\rho ^t~a.s. \end{aligned}$$
Proof
From \((\mathrm{A}1),\) we have
$$\begin{aligned} |\lambda _t(\theta ^*)-{\tilde{\lambda }}_t(\theta ^*)|\le k_2^{t-1}|\lambda _1(\theta ^*)-{\tilde{\lambda }}_1(\theta ^*)|, \end{aligned}$$
hence, the lemma is validated by \((\mathrm{A}3)\). \(\square \)
Lemma A.2
Under \((\mathrm{A}1){-}(\mathrm{A}3)\), we have
$$\begin{aligned} \sup _{\theta \in \Theta }\bigg |\frac{1}{n}\sum _{t=1}^{n}l_{\alpha ,t} (\theta )-\frac{1}{n}\sum _{t=1}^{n}\tilde{l}_{\alpha ,t}(\theta )\bigg |\rightarrow 0~a.s. \end{aligned}$$
Proof
It suffices to show that
$$\begin{aligned} \sup _{\theta \in \Theta }\left| l_{\alpha ,t}(\theta )-\tilde{l}_{\alpha ,t}(\theta )\right| \rightarrow 0~a.s. \end{aligned}$$
Note that \(\left| l_{\alpha ,t}(\theta )-\tilde{l}_{\alpha ,t}(\theta )\right| \le W_{1t}(\theta )+W_{2t}(\theta )\), where
$$\begin{aligned} W_{1t}(\theta )&=|m_{\alpha ,t,1}(\theta )-\tilde{m}_{\alpha ,t,1}(\theta )|\\&=\left| \sum \limits _{x=0}^{\infty }P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})-\sum \limits _{x=0}^{\infty }\tilde{P}_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\right| ,\\ W_{2t}(\theta )&=\left( 1+\frac{1}{\alpha }\right) |m_{\alpha ,t,2}(\theta )-\tilde{m}_{\alpha ,t,2}(\theta )|\\&=\left( 1+\frac{1}{\alpha }\right) \left| P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})-\tilde{P}_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\right| . \end{aligned}$$
According to the mean value theorem (MVT) and (P1)–(P3), it holds that
$$\begin{aligned} W_{1t}(\theta )&=(1+\alpha )|\lambda _t(\theta ^*)-{\tilde{\lambda }}_t(\theta ^*)|\left| \sum \limits _{x=0}^{\infty }\bar{P}_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\frac{1}{1+\sigma \bar{\lambda }_t(\theta ^*)}\left( \frac{x}{\bar{\lambda }_t(\theta ^*)}-1\right) \right| \\&\le 2(1+\alpha )|\lambda _t(\theta ^*)-{\tilde{\lambda }}_t(\theta ^*)|, \end{aligned}$$
for some intermediate point \(\bar{\lambda }_t(\theta ^*)\) between \(\lambda _t(\theta ^*)\) and \({\tilde{\lambda }}_t(\theta ^*)\), and \(\bar{P}_{\theta }(x|\mathcal {F}_{t-1})\) is defined by substituting \(\lambda _t(\theta ^*)\) with \(\bar{\lambda }_t(\theta ^*)\) in \(P_{\theta }(x|\mathcal {F}_{t-1})\). Hence,
$$\begin{aligned} \sup _{\theta \in \Theta }W_{1t}\le 2(1+\alpha )V\rho ^t\xrightarrow {a.s.}0~as~t\rightarrow \infty . \end{aligned}$$
Next, note that
$$\begin{aligned} W_{2t}(\theta )&=(1+\alpha )|\lambda _t(\theta ^*)-{\tilde{\lambda }}_t (\theta ^*)|\left| \sum \limits _{x=0}^{\infty }\bar{P}_{\theta }^{\alpha } (X_t|\mathcal {F}_{t-1})\frac{1}{1+\sigma \bar{\lambda }_t(\theta ^*)} \left( \frac{X_t}{\bar{\lambda }_t(\theta ^*)}-1\right) \right| \\&\le (1+\alpha )\left( \frac{X_t}{\delta _{L}}+1\right) |\lambda _t (\theta ^*)-{\tilde{\lambda }}_t(\theta ^*)|. \end{aligned}$$
Thus,
$$\begin{aligned} \sup _{\theta \in \Theta }W_{2t}\le (1+\alpha ) \left( \frac{X_t}{\delta _{L}}+1\right) V\rho ^t, \end{aligned}$$
where \(\bar{P}_{\theta }(X_t|\mathcal {F}_{t-1})\) is defined by substituting x with \(X_t\) in \(\bar{P}_{\theta }(x|\mathcal {F}_{t-1})\). Since \(X_t\rho ^t\rightarrow 0~a.s.\) as \(t\rightarrow \infty \) (see Proposition 1 of Francq and Zakoïan [14]), \(\sup _{\theta \in \Theta }W_{2t}\) converges to 0 a.s. Therefore, the lemma is validated. \(\square \)
Lemma A.3
Under \((\mathrm{A}1){-}(\mathrm{A}4)\), we have
$$\begin{aligned} E\left( \sup _{\theta \in \Theta } l_{\alpha ,t}(\theta )\right) <\infty ~~and~~if~~ \theta \ne \theta _0,~~then~~ El_{\alpha ,t}(\theta )>E l_{\alpha ,t}(\theta _0). \end{aligned}$$
Proof
From \((\mathrm{P}1)\) and \((\mathrm{P}2)\), we obtain
$$\begin{aligned} \left| l_{\alpha ,t}(\theta )\right|&\le m_{\alpha ,t,1}(\theta )+\left( 1+\frac{1}{\alpha }\right) m_{\alpha ,t,2}(\theta )\\&=\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})+\left( 1+\frac{1}{\alpha }\right) P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1}) \le 2+\frac{1}{\alpha }, \end{aligned}$$
and thus the first part of the lemma is established.
Note that,
$$\begin{aligned}&E l_{\alpha ,t}(\theta )-E l_{\alpha ,t}(\theta _0)&\\&\quad =E\left[ E(l_{\alpha ,t}(\theta )-l_{\alpha ,t}(\theta _0)|\mathcal {F}_{t-1})\right] \\&\quad =E \left[ \sum _{x=0}^{\infty }\left( P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})-\left( 1+\frac{1}{\alpha }\right) P_{\theta }^{\alpha }(x|\mathcal {F}_{t-1})P_{\theta _0}(x|\mathcal {F}_{t-1})\right. \right. \\&\left. \left. \qquad +\frac{1}{\alpha }P_{\theta _0}^{1+\alpha }(x|\mathcal {F}_{t-1})\right) \right] \ge 0, \end{aligned}$$
where the equality holds if and only if \(\sigma =\sigma _0\) and \(\lambda _t(\theta ^*)=\lambda _t(\theta _0^*)~a.s.\) Therefore, by (A4) the lemma is proved. \(\square \)
Proof of Theorem 3.4
Express
$$\begin{aligned} \sup _{\theta \in \Theta }\left| \frac{1}{n}\sum _{t=1}^{n}\tilde{l}_{\alpha ,t}(\theta )-E l_{\alpha ,t}(\theta )\right|\le & {} \sup _{\theta \in \Theta }\left| \frac{1}{n}\sum _{t=1}^{n}\tilde{l}_{\alpha ,t} (\theta )-\frac{1}{n}\sum _{t=1}^{n}l_{\alpha ,t}(\theta )\right| \\&+\,\sup _{\theta \in \Theta }\left| \frac{1}{n}\sum _{t=1}^{n}l_{\alpha ,t} (\theta )-E l_{\alpha ,t}(\theta )\right| . \end{aligned}$$
The first term of the above inequality converges to 0 a.s. by Lemma lemma2A.2A. Since \(l_{\alpha ,t}(\theta )\) is stationary and ergodic and \(E \left( \sup _{\theta \in \Theta }l_{\alpha ,t}(\theta )\right) <\infty \), the second term also converges to 0 a.s. (see Theorem 2.7 of Straumann and Mikosch [27]). Further, \(E l_{\alpha ,t}(\theta )\) has a unique minimum at \(\theta _0\), therefore, the theorem is established.
Let
$$\begin{aligned} K_{\alpha ,t,1}(x;\theta )=\frac{\Gamma ^{'}(x+1/\sigma )}{\Gamma (1/\sigma )} -\frac{\Gamma ^{'}(1/\sigma )}{\Gamma (1/\sigma )} +\frac{\lambda _t(\theta ^*)-x}{\lambda _t(\theta ^*)+1/\sigma } -\ln (\sigma \lambda _t(\theta ^*)+1) \end{aligned}$$
with
$$\begin{aligned} \frac{\Gamma ^{'}(x+1/\sigma )}{\Gamma (1/\sigma )}-\frac{\Gamma ^{'}(1/\sigma )}{\Gamma (1/\sigma )}= {\left\{ \begin{array}{ll} \frac{1}{1/\sigma +x-1}+ \frac{1}{1/\sigma +x-2}+\cdots +\frac{1}{1/\sigma },&{}x=1,2,\ldots ,\\ 0,&{}x=0, \end{array}\right. } \end{aligned}$$
and
$$\begin{aligned} K_{\alpha ,t,2}(x;\theta )= & {} \frac{\partial K_{\alpha ,t,1}(x;\theta )}{\partial (1/\sigma )}\\= & {} {\left\{ \begin{array}{ll} -\frac{1}{(1/\sigma +x-1)^2} -\frac{1}{(1/\sigma +x-2)^2} -\cdots -\frac{1}{(1/\sigma )^2}\\ -\frac{\lambda _t(\theta ^*)-x}{(1/\sigma +\lambda _t(\theta ^*))^2} +\frac{\lambda _t(\theta ^*)}{1/\sigma (1/\sigma +\lambda _t(\theta ^*))}, &{}x=1,2,\ldots ,\\ \frac{-\lambda _t(\theta ^*)}{(1/\sigma +\lambda _t(\theta ^*))^2} +\frac{\lambda _t(\theta ^*)}{1/\sigma (1/\sigma +\lambda _t(\theta ^*))},&{}x=0. \end{array}\right. } \end{aligned}$$
\(K_{\alpha ,t,1}(X_t;\theta )\) and \(K_{\alpha ,t,1}(X_t;\theta )\) are the counterparts of \(K_{\alpha ,t,1}(x;\theta )\) and \(K_{\alpha ,t,1}(x;\theta )\) substituting x with \(X_t\). Now, we derive the first and second derivatives of \(l_{\alpha ,t}(\theta )\). The first derivatives are as follows:
$$\begin{aligned} \frac{\partial l_{\alpha ,t}(\theta )}{\partial \theta }=\frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta }-\left( 1+\frac{1}{\alpha }\right) \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta } \end{aligned}$$
with
$$\begin{aligned} \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \sigma }&=-\frac{1+\alpha }{\sigma ^2}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})K_{\alpha ,t,1}(x;\theta ),\\ \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta ^*}&=\frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)} \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\\&=(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\frac{1}{\sigma \lambda _t(\theta )^*}\left( \frac{x}{\lambda _t(\theta ^*)}-1\right) \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*},\\ \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \sigma }&=-\frac{\alpha }{\sigma ^2} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})K_{\alpha ,t,1}(X_t;\theta ),\\ \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta ^*}&=\frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)} \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\\&=\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\frac{1}{\sigma \lambda _t(\theta )^*} \left( \frac{X_t}{\lambda _t(\theta ^*)}-1\right) \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}. \end{aligned}$$
The second derivatives which will be used in Lemmas A–A are obtained as follows:
$$\begin{aligned} \frac{\partial ^2 l_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^{\top }}=\frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \theta \partial \theta ^{\top }}-\left( 1+\frac{1}{\alpha }\right) \frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \theta \partial \theta ^{\top }} \end{aligned}$$
with
$$\begin{aligned}&\frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \sigma ^2}=\frac{2(1+\alpha )}{\sigma ^3} \sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})K_{\alpha ,t,1}(x;\theta )-\frac{(1+\alpha )}{\sigma ^2}\\&\qquad \left[ -\frac{1+\alpha }{\sigma ^2} \sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})K_{\alpha ,t,1}^2(x;\theta ) -\frac{1}{\sigma ^2}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})K_{\alpha ,t,2}(x;\theta )\right] ,\\&\frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \theta ^*\partial {\theta ^*}^{\top }} =\frac{\partial ^2m_{\alpha ,t,1}(\theta )}{\partial \lambda _t^2(\theta ^*)} \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}+\frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)} \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\\&\qquad =(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left[ (1+\alpha )\left( \frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \frac{x}{\lambda _t(\theta ^*)}-1\right) \right) ^2 \right. \\&\qquad \left. -\,\frac{1}{(\sigma \lambda _t(\theta ^*)+1)^2}\left( \frac{x}{\lambda _t(\theta ^*)}-1\right) -\frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \frac{x}{\lambda _t(\theta ^*)^2} \right) \right] \frac{\partial \lambda _t (\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}\\&\qquad +\,(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1}) \frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \frac{x}{\lambda _t(\theta ^*)}-1\right) \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }},\\&\frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \sigma \partial \theta ^*} =\frac{\partial (\partial m_{\alpha ,t,1}(\theta )/\partial \sigma )}{\partial \theta ^*}=\frac{\partial (\partial m_{\alpha ,t,1}(\theta )/\partial \sigma )}{\partial \lambda _t(\theta ^*)} \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\\&\quad =\left[ -\frac{(1+\alpha )^2}{\sigma ^2}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha } (x|\mathcal {F}_{t-1})K_{\alpha ,t,1}(x;\theta )\frac{1}{\sigma \lambda _t(\theta ^*)+1}\right. \\&\qquad \quad \left. \left( \frac{x}{\lambda _t(\theta ^*)}-1\right) +(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left( \frac{x}{\lambda _t(\theta ^*)}-1\right) \right. \\&\qquad \quad \quad \left. \frac{-\lambda _t(\theta ^*)}{(\sigma \lambda _t(\theta ^*)+1)^2}\right] \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*},\\&\frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \sigma ^2}=\frac{2\alpha }{\sigma ^3} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})K_{\alpha ,t,1}(X_t;\theta )-\frac{\alpha }{\sigma ^2}\\&\qquad \quad \left[ -\frac{\alpha }{\sigma ^2} P_{\theta }^{\alpha }(X_t| \mathcal {F}_{t-1})K_{\alpha ,t,1}^2(X_t,\theta )-\frac{1}{\sigma ^2} P_{\theta }^{\alpha } (X_t|\mathcal {F}_{t-1})K_{\alpha ,t,2}(X_t;\theta )\right] ,\\&\frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \theta ^*\partial {\theta ^*}^{\top }}=\frac{\partial ^2m_{\alpha ,t,2}(\theta )}{\partial \lambda _t^2 (\theta ^*)}\frac{\partial \lambda _t (\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}+\frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)} \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\\&\quad =\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left[ \alpha \left( \frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \frac{X_t}{\lambda _t(\theta ^*)}-1\right) \right) ^2\right. \\&\qquad \left. -\,\frac{1}{(\sigma \lambda _t(\theta ^*)+1)^2}\left( \frac{X_t}{\lambda _t(\theta ^*)} -1\right) -\frac{1}{\sigma \lambda _t(\theta ^*)+1}\frac{X_t}{\lambda _t(\theta ^*)^2}\right] \\&\qquad \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}+\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\frac{1}{\sigma \lambda _t(\theta ^*) +1}\left( \frac{X_t}{\lambda _t(\theta ^*)}-1\right) \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }},\\&\frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \sigma \partial \theta ^*} =\frac{\partial (\partial m_{\alpha ,t,2}(\theta )/\partial \sigma )}{\partial \theta ^*} =\frac{\partial (\partial m_{\alpha ,t,2}(\theta )/\partial \sigma )}{\partial \lambda _t (\theta ^*)}\frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\\&\quad =\left[ -\frac{\alpha ^2}{\sigma ^2} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1}) K_{\alpha ,t,1}(X_t;\theta )\frac{1}{\sigma \lambda _t(\theta ^*)+1}\right. \left( \frac{X_t}{\lambda _t(\theta ^*)}-1\right) \\&\qquad \left. +\,\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left( \frac{X_t}{\lambda _t (\theta ^*)}-1\right) \frac{-\lambda _t(\theta ^*)}{(\sigma \lambda _t(\theta ^*)+1)^2}\right] \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}. \end{aligned}$$
\(\square \)
Lemma A.4
Under(A1)-(A3) and (P1)-(P4), we have
$$\begin{aligned} |K_{\alpha ,t,1}(x;\theta )|&\le \frac{x}{1/\sigma }+\frac{x}{1/\sigma +\lambda _t(\theta ^*)}+1+\sigma \lambda _t(\theta ^*),\\ |K_{\alpha ,t,2}(x;\theta )|&\le \frac{x}{(1/\sigma )^2}+\frac{\lambda _t(\theta ^*)}{(1/\sigma +\lambda _t(\theta ^*))^2}+\frac{x}{(1/\sigma +\lambda _t(\theta ^*))^2}+\sigma ,\\ \left| K_{\alpha ,t,1}(X_t;\theta )\right|&\le \frac{X_t}{1/\sigma }+\frac{X_t}{1/\sigma +\lambda _t(\theta ^*)}+1+\sigma \lambda _t(\theta ^*),\\ \left| K_{\alpha ,t,2}(X_t;\theta )\right|&\le \frac{X_t}{(1/\sigma )^2}+\frac{\lambda _t(\theta ^*)}{(1/\sigma +\lambda _t(\theta ^*))^2}+\frac{X_t}{(1/\sigma +\lambda _t(\theta ^*))^2}+\sigma ,\\ \left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \sigma }\right|&\le \frac{1+\alpha }{\sigma ^2}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(x;\theta )\right| \\&\le \frac{1+\alpha }{\sigma ^2}\left( \frac{\lambda _t(\theta ^*)}{(1/\sigma )^2}+\sigma \lambda _t(\theta ^*) +2\right) \\&\quad =(1+\alpha )\left( 1+\frac{1}{\sigma }\right) \lambda _t(\theta ^*)+\frac{2(1+\alpha )}{\sigma ^2},\\ \left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}\right|&\le \frac{1+\alpha }{\sigma \lambda _t(\theta ^*)+1}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left( \frac{x}{\lambda _t(\theta ^*)}+1\right) \le \frac{2(1+\alpha )}{\sigma \lambda _t(\theta ^*)+1}\\&\quad \le 2(1+\alpha ),\\ \left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \sigma }\right|&\le \frac{\alpha }{\sigma ^2} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(X_t;\theta )\right| \le \frac{2\alpha }{\sigma }X_t+\frac{\alpha }{\sigma }\lambda _t(\theta ^*)+\frac{\alpha }{\sigma ^2},\\ \left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}\right|&\le \frac{\alpha }{\sigma \lambda _t(\theta ^*)+1} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left( \frac{X_t}{\lambda _t(\theta ^*)}+1\right) \le \frac{\alpha }{\lambda _t(\theta ^*)}X_t+\alpha ,\\ \left| \frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \sigma ^2}\right|&\le \frac{2(1+\alpha )}{\sigma ^3} \sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(x;\theta )\right| \\&\quad +\frac{(1+\alpha )^2}{\sigma ^4}\sum _{x=0}^{\infty }P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(x;\theta )\right| ^2\\&\quad +\frac{1+\alpha }{\sigma ^4}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left| K_{\alpha ,t,2}(x;\theta )\right| \\&\le \frac{4(1+\alpha )}{\sigma ^2}\lambda _t(\theta *)+\frac{4(1+\alpha )}{\sigma ^3}+\frac{2(1+\alpha )^2(\sigma +2)}{\sigma ^2}\lambda _t^2(\theta *)\\&\quad +\frac{2(1+\alpha )^2}{\sigma ^2}\lambda _t(\theta *)+\frac{2(1+\alpha )^2}{\sigma ^3}+\frac{4(1+\alpha )^2}{\sigma ^4}+\frac{2(1+\alpha )^2}{\sigma ^4\lambda _t(\theta *)}\\&\quad +\frac{1+\alpha }{\sigma ^2}\lambda _t(\theta *)+\frac{1+\alpha }{\sigma ^3}+\frac{2(1+\alpha )}{\sigma ^4\lambda _t(\theta *)}\\&=2(1+\alpha )^2\left( \frac{1}{\sigma }+\frac{2}{\sigma ^2}\right) \lambda _t^2(\theta *)+\frac{(1+\alpha )(2\alpha +7)}{\sigma ^2}\lambda _t(\theta *)\\&\quad +\frac{(1+\alpha )(2\alpha +7)}{\sigma ^3}+\frac{4(1+\alpha )^2}{\sigma ^4}+\frac{(1+\alpha )(2\alpha +4)}{\sigma ^4\lambda _t(\theta *)},\\ \left| \frac{\partial ^2m_{\alpha ,t,1}(\theta )}{\partial \lambda _t^2(\theta ^*)}\right|&\le (1+\alpha )^2\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \left( \frac{x}{\lambda _t(\theta ^*)}\right) ^2+1\right) \\&\quad +(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\frac{1}{(\sigma \lambda _t(\theta ^*)+1)^2}\left( \frac{x}{\lambda _t(\theta ^*)}+1\right) \\&\quad +(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\frac{1}{\sigma \lambda _t(\theta ^*)+1}\frac{x}{\lambda _t^2(\theta ^*)}\\&\le (1+\alpha )^2\left( \frac{1}{\lambda _t(\theta ^*)}+2\right) +2(1+\alpha )+\frac{1+\alpha }{\lambda _t(\theta ^*)}\\&=2(1+\alpha )(\alpha +2)+\frac{(1+\alpha )(\alpha +2)}{\lambda _t(\theta ^*)},\\ \left| \frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \sigma \partial \lambda _t(\theta ^*)}\right|&\le \frac{(1+\alpha )^2}{\sigma ^2}\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(x;\theta )\right| \frac{1}{\sigma \lambda _t(\theta ^*)+1}\\&\quad \left( \frac{x}{\lambda _t(\theta ^*)}+1\right) \\&\quad +(1+\alpha )\sum _{x=0}^{\infty } P_{\theta }^{1+\alpha }(x|\mathcal {F}_{t-1})\left( \frac{x}{\lambda _t(\theta ^*)}+1\right) \frac{\lambda _t(\theta ^*)}{(\sigma \lambda _t(\theta ^*)+1)^2}\\&\le \frac{(1+\alpha )^2}{\sigma ^2}\left( 3\sigma \lambda _t(\theta ^*)+\sigma +\frac{1}{1/\sigma +\lambda _t(\theta ^*)}+5\right) +2(1+\alpha )\lambda _t(\theta ^*)\\&\le \left( \frac{3(1+\alpha )^2}{\sigma ^2}+2(1+\alpha )\right) \lambda _t(\theta ^*)+\frac{2(1+\alpha )^2}{\sigma }+\frac{5(1+\alpha )^2}{\sigma ^2},\\ \left| \frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \sigma ^2}\right|&\le \frac{2\alpha }{\sigma ^3} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(X_t;\theta )\right| +\frac{\alpha ^2}{\sigma ^4}P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\\&\quad \left| K_{\alpha ,t,1}(X_t;\theta )\right| ^2\\&\quad +\frac{\alpha }{\sigma ^4} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left| K_{\alpha ,t,2}(X_t;\theta )\right| \\&\le \left( \frac{4\alpha }{\sigma ^2}X_t+\frac{2\alpha }{\sigma ^2}\lambda _t(\theta *)+\frac{2\alpha }{\sigma ^3}\right) +\left( \frac{4\alpha ^2}{\sigma ^2}X_t^2+\frac{2\alpha ^2}{\sigma ^2}\lambda _t^2(\theta *)+\frac{2\alpha ^2}{\sigma ^4}\right) \\&\quad +\left( \frac{2\alpha }{\sigma ^2}X_t+\frac{\alpha }{\sigma ^3}+\frac{\alpha }{\sigma ^4\lambda _t(\theta *)}\right) \\&=\frac{4\alpha ^2}{\sigma ^2}X_t^2+\frac{6\alpha }{\sigma ^2}X_t+\frac{2\alpha ^2}{\sigma ^2}\lambda _t^2(\theta *)+\frac{2\alpha }{\sigma ^2}\lambda _t(\theta *)+\frac{3\alpha }{\sigma ^3}\\&\quad +\frac{2\alpha ^2}{\sigma ^4}+\frac{\alpha }{\sigma ^4\lambda _t(\theta *)},\\ \left| \frac{\partial ^2m_{\alpha ,t,2}(\theta )}{\partial \lambda _t^2(\theta ^*)}\right|&\le \alpha ^2 P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \left( \frac{X_t}{\lambda _t(\theta ^*)}\right) ^2+1\right) \\&\quad +\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\frac{1}{(\sigma \lambda _t(\theta ^*)+1)^2}\left( \frac{X_t}{\lambda _t(\theta ^*)}+1\right) +\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\\&\quad \frac{1}{\sigma \lambda _t(\theta ^*)+1}\frac{X_t}{\lambda _t^2(\theta ^*)}\\&\le \left( \frac{\alpha ^2}{\lambda _t^2(\theta ^*)}X_t^2+\alpha ^2\right) +\left( \frac{\alpha }{\lambda _t(\theta ^*)}X_t+\alpha \right) +\left( \frac{\alpha }{\lambda _t^2(\theta ^*)}X_t\right) \\&=\frac{\alpha ^2}{\lambda _t^2(\theta ^*)}X_t^2+\left( \frac{\alpha }{\lambda _t(\theta ^*)}+\frac{\alpha }{\lambda _t^2(\theta ^*)}\right) X_t+\alpha ^2+\alpha ,\\ \left| \frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \sigma \partial \lambda _t(\theta ^*)}\right|&\le \frac{\alpha ^2}{\sigma ^2} P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left| K_{\alpha ,t,1}(X_t;\theta )\right| \frac{1}{\sigma \lambda _t(\theta ^*)+1}\left( \frac{X_t}{\lambda _t(\theta ^*)}+1\right) \\&\quad +\alpha P_{\theta }^{\alpha }(X_t|\mathcal {F}_{t-1})\left( \frac{X_t}{\lambda _t(\theta ^*)}+1\right) \frac{\lambda _t(\theta ^*)}{(\sigma \lambda _t(\theta ^*)+1)^2}\\&\le \frac{\alpha ^2}{\sigma ^2}\left( \frac{2X_t^2}{1/\sigma \lambda _t(\theta ^*)}+3\sigma X_t+\frac{X_t}{\lambda _t(\theta ^*)}+\sigma \lambda _t(\theta ^*)+1\right) \\&\quad +\alpha \left( \frac{X_t}{\lambda _t(\theta ^*)}+1\right) \lambda _t(\theta ^*)\\&=\frac{2\alpha ^2}{\sigma \lambda _t(\theta ^*)}X_t^2+\left( \alpha +\frac{3\alpha ^2}{\sigma }+\frac{\alpha ^2}{\sigma ^2\lambda _t(\theta ^*)}\right) X_t+\left( \alpha +\frac{\alpha ^2}{\sigma }\right) \lambda _t(\theta ^*)\\&\quad +\frac{\alpha ^2}{\sigma ^2}. \end{aligned}$$
Note that owing to the MVT, we get
$$\begin{aligned} \left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \sigma }-\frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \sigma }\right|&\le \left| \frac{\partial (\partial m_{\alpha ,t,1}(\theta )/\partial \sigma )}{\partial \lambda _t(\theta ^*)}\right| _{\lambda _t(\theta ^*)=\bar{\lambda }_t(\theta ^*)}\left| \lambda _t(\theta ^*)-{\tilde{\lambda }}_t(\theta ^*)\right| \\&\le \left( \left( \frac{3(1+\alpha )^2}{\sigma ^2}+2(1+\alpha )\right) \right. \\&\quad \left. \bar{\lambda }_t(\theta ^*)+\frac{2(1+\alpha )^2}{\sigma }+\frac{5(1+\alpha )^2}{\sigma ^2}\right) \\&\quad V\rho ^t,\\ \left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}-\frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}\right|&\le \left| \frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \lambda _t^2(\theta ^*)}\right| _{\lambda _t(\theta ^*)=\bar{\lambda }_t(\theta ^*)}\left| \lambda _t(\theta ^*)-\tilde{\lambda }_t(\theta ^*)\right| \\&\le \left( 2(1+\alpha )(\alpha +2)+\frac{(1+\alpha )(\alpha +2)}{\bar{\lambda }_t(\theta ^*)}\right) V\rho ^t,\\ \left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \sigma }-\frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \sigma }\right|&\le \left| \frac{\partial (\partial m_{\alpha ,t,2}(\theta )/\partial \sigma )}{\partial \lambda _t(\theta ^*)}\right| _{\lambda _t(\theta ^*)=\bar{\lambda }_t(\theta ^*)}\left| \lambda _t(\theta ^*)-\tilde{\lambda }_t(\theta ^*)\right| \\&\le \left( \frac{2\alpha ^2}{\sigma \bar{\lambda }_t(\theta ^*)}X_t^2+\left( \alpha +\frac{3\alpha ^2}{\sigma }+\frac{\alpha ^2}{\sigma ^2\bar{\lambda }_t(\theta ^*)}\right) X_t\right. \\&\quad \left. +\left( \alpha +\frac{\alpha ^2}{\sigma }\right) \bar{\lambda }_t(\theta ^*) +\frac{\alpha ^2}{\sigma ^2}\right) V\rho ^t,\\ \left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}-\frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}\right|&\le \left| \frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \lambda _t^2(\theta ^*)}\right| _{\lambda _t(\theta ^*)=\bar{\lambda }_t(\theta ^*)}\left| \lambda _t(\theta ^*)-\tilde{\lambda }_t(\theta ^*)\right| \\&\le \left( \frac{\alpha ^2}{\bar{\lambda }_t^2(\theta ^*)}X_t^2+\left( \frac{\alpha }{\bar{\lambda }_t(\theta ^*)}+\frac{\alpha }{\bar{\lambda }_t^2(\theta ^*)}\right) X_t+\alpha ^2+\alpha \right) \\&\quad \quad V\rho ^t. \end{aligned}$$
Lemma A.5
Under \((\mathrm{A}1){-}(\mathrm{A}6)\), we have
$$\begin{aligned} E\left( \sup _{\theta \in \Theta }\left\| \dfrac{\partial ^2l_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^{\top }}\right\| \right)<\infty ~~and~~ E\left( \sup _{\theta \in \Theta }\left\| \dfrac{\partial l_{\alpha ,t}(\theta )}{\partial \theta }\dfrac{\partial l_{\alpha ,t}(\theta )}{\partial \theta ^{\top }} \right\| \right) <\infty . \end{aligned}$$
Proof
For the first part, it suffices to show that \(E\left( \sup \limits _{\theta \in \Theta }\left\| \dfrac{\partial ^2m_{\alpha ,t,1}(\theta )}{\partial \theta \partial \theta ^{\top }}\right\| \right) <\infty \) and \(E\left( \sup \limits _{\theta \in \Theta }\left\| \dfrac{\partial ^2m_{\alpha ,t,2}(\theta )}{\partial \theta \partial \theta ^{\top }}\right\| \right) <\infty \). By using (A2) and Lemma A, we have
$$\begin{aligned} \sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,1}^2(\theta )}{\partial \sigma ^2}\right|&\le 2(1+\alpha )^2\left( \frac{1}{\delta _{L}}+\frac{2}{\delta _{L}^2}\right) \sup _{\theta ^*\in \Theta ^*}\lambda _t^2(\theta ^*)\\&\quad +\,\frac{(1+\alpha )(2\alpha +7)}{\delta _{L}^2}\sup _{\theta ^*\in \Theta ^*}\lambda _t(\theta ^*)+\frac{(1+\alpha )(2\alpha +7)}{\delta _{L}^3}\\&\quad +\,\frac{4(1+\alpha )^2}{\delta _{L}^4}+\frac{(1+\alpha )(2\alpha +4)}{\delta _{L}^5},\\ \sup _{\theta \in \Theta }\left\| \frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \theta ^*\partial {\theta ^*}^{\top }}\right\|&\le \sup _{\theta \in \Theta }\left| \frac{\partial ^2m_{\alpha ,t,1}(\theta )}{\partial \lambda _t^2(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}\right\| \\&\quad +\,\sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\right\| \\&\le \left( 2(1+\alpha )(\alpha +2)+\frac{(1+\alpha )(\alpha +2)}{\delta _{L}}\right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}\right\| \\&+2(1+\alpha )\sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\right\| ,\\ \sup _{\theta \in \Theta }\left\| \frac{\partial ^2 m_{\alpha ,t,1}(\theta )}{\partial \sigma \partial \theta ^*}\right\|&\le \sup _{\theta \in \Theta }\left| \frac{\partial (\partial m_{\alpha ,t,1}(\theta )/\partial \sigma )}{\partial \lambda _t(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \\&\le \left( \left( \frac{3(1+\alpha )^2}{\delta _{L}^2}+2(1+\alpha )\right) \sup _{\theta ^*\in \Theta ^*}\lambda _t(\theta ^*)+\frac{2(1+\alpha )^2}{\delta _{L}}\right. \\&\quad +\left. \,\frac{5(1+\alpha )^2}{\delta _{L}^2}\right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| ,\\ \sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,2}^2(\theta )}{\partial \sigma ^2}\right|&\le \frac{4\alpha ^2}{\delta _{L}^2}X_t^2+\frac{6\alpha }{\delta _{L}^2}X_t+\frac{2\alpha ^2}{\delta _{L}^2}\sup _{\theta ^*\in \Theta ^*}\lambda _t^2(\theta ^*)\\&\quad +\,\frac{2\alpha }{\delta _{L}^2}\sup _{\theta ^*\in \Theta ^*}\lambda _t(\theta ^*)+\frac{3\alpha }{\delta _{L}^3}+\frac{2\alpha ^2}{\delta _{L}^4}+\frac{\alpha }{\delta _{L}^5},\\ \sup _{\theta \in \Theta }\left\| \frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \theta ^*\partial {\theta ^*}^{\top }}\right\|&\le \sup _{\theta \in \Theta }\left| \frac{\partial ^2m_{\alpha ,t,2}(\theta )}{\partial \lambda _t^2(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}\right\| \\&\quad +\,\sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\right\| \\&\le \left( \frac{\alpha ^2}{\delta _{L}^2}X_t^2+\left( \frac{\alpha }{\delta _{L}}+\frac{\alpha }{\delta _{L}^2}\right) X_t+\alpha ^2+\alpha \right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\frac{\partial \lambda _t(\theta ^*)}{\partial {\theta ^*}^{\top }}\right\| \\&+\left( \frac{\alpha }{\delta _{L}}X_t+\alpha \right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial ^2\lambda _t(\theta ^*)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\right\| ,\\ \sup _{\theta \in \Theta }\left\| \frac{\partial ^2 m_{\alpha ,t,2}(\theta )}{\partial \sigma \partial \theta ^*}\right\|&\le \sup _{\theta \in \Theta }\left| \frac{\partial (\partial m_{\alpha ,t,1}(\theta )/\partial \sigma )}{\partial \lambda _t(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \\&\le \left( \frac{2\alpha ^2}{\delta _{L}^2}X_t^2+\left( \alpha +\frac{3\alpha ^2}{\delta _{L}}+\frac{\alpha ^2}{\delta _{L}^3}\right) X_t\right. \\&\quad +\,\left. \left( \alpha +\frac{\alpha ^2}{\delta _{L}}\right) \sup _{\theta ^*\in \Theta ^*}\lambda _t(\theta ^*)+\frac{\alpha ^2}{\delta _{L}^2}\right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| . \end{aligned}$$
Hence, the first part of the lemma is verified by the Cauchy–Schwarz inequality, (A3), (A6), and the fact that \(X_t\) has finite moments of any order. For the second part, it is sufficient to show that
$$\begin{aligned} E\left( \sup _{\theta \in \Theta }\left\| \dfrac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta }\dfrac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta ^{\top }}\right\| \right)<\infty , ~~E\left( \sup _{\theta \in \Theta }\left\| \dfrac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta }\dfrac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta ^{\top }}\right\| \right) <\infty \end{aligned}$$
and
$$\begin{aligned} E\left( \sup _{\theta \in \Theta }\left\| \dfrac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta }\dfrac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta ^{\top }}\right\| \right) <\infty . \end{aligned}$$
In a similar way to the proof of the first part, from (A2), (P1)–(P3) and Lemma lemma4A.4A, we can show that
$$\begin{aligned} \sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \sigma }\right|&\le (1+\alpha )\left( 1+\frac{1}{\delta _{L}}\right) \sup _{\theta ^*\in \Theta ^*}\lambda _t(\theta ^*)+\frac{2(1+\alpha )}{\delta _{L}^2},\\ \sup _{\theta \in \Theta }\left\| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta ^*}\right\|&\le \left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \le 2(1+\alpha )\sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| ,\\ \sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \sigma }\right|&\le \frac{2\alpha }{\delta _{L}}X_t+\frac{\alpha }{\delta _{L}}\sup _{\theta ^*\in \Theta ^*}\lambda _t(\theta ^*)+\frac{\alpha }{\delta _{L}^2},\\ \sup _{\theta \in \Theta }\left\| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta ^*}\right\|&\le \left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}\right| \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \\&\le \left( \frac{\alpha }{\delta _{L}}X_t+\alpha \right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| . \end{aligned}$$
\(\square \)
The product of every pair of above has finite expectations because of the Cauchy–Schwarz inequality, (A3), (A6) and the existence of any order moments of \(X_t\), the second part is validated. Thus the lemma is established.
Lemma A.6
Under \((\mathrm{A}1){-}(\mathrm{A}7)\), we have
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial l_{\alpha ,t}(\theta )}{\partial \theta }-\frac{\partial \tilde{l}_{\alpha ,t}(\theta )}{\partial \theta }\right\| \xrightarrow {a.s.}0~~as~~n \rightarrow \infty . \end{aligned}$$
Proof
It suffices to show that
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta }-\frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \theta ^{\top }}\right\| \xrightarrow {a.s.}0 \end{aligned}$$
and
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta }-\frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \theta ^{\top }} \right\| \xrightarrow {a.s.}0~~as~~n\rightarrow \infty . \end{aligned}$$
From Lemma lemma4A.4A, we have
$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \sigma }-\frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \sigma }\right| \\&\quad \le \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\left( \left( \frac{3(1+\alpha )^2}{\delta _{L}^2}+2(1+\alpha )\right) \sup _{\theta ^*\in \Theta ^*}\bar{\lambda }_t(\theta ^*)\right. \\&\qquad \left. +\frac{2(1+\alpha )^2}{\delta _{L}}+\frac{5(1+\alpha )^2}{\delta _{L}^2}\right) V\rho ^t,\\&\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \theta }-\frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \theta }\right\| \\&\quad \le \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \left( \frac{\partial m_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}-\frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}\right) \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \\&\qquad +\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial \tilde{m}_{\alpha ,t,1}(\theta )}{\partial \lambda _t(\theta ^*)}\left( \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}-\frac{\partial \tilde{\lambda }_t(\theta ^*)}{\partial \theta ^*}\right) \right\| \\&\quad \le \left( \left( 2(1+\alpha )(\alpha +2)+\frac{(1+\alpha )(\alpha +2)}{\delta _{L}}\right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \right. \\&\qquad \left. +2(1+\alpha )\right) V\rho ^t,\\&\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \sigma }-\frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \sigma }\right| \\&\quad \le \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\left( \frac{2\alpha ^2}{\delta _{L}^2}X_t^2+\left( \alpha +\frac{3\alpha ^2}{\delta _{L}}+\frac{\alpha ^2}{\delta _{L}^3}\right) X_t\right. \\&\qquad \left. +\,\left( \alpha +\frac{\alpha ^2}{\delta _{L}}\right) \sup _{\theta ^*\in \Theta ^*}\bar{\lambda }_t(\theta ^*)+\frac{\alpha ^2}{\delta _{L}^2}\right) V\rho ^t,\\&\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \theta }-\frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \theta }\right\| \\&\quad \le \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \left( \frac{\partial m_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}-\frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}\right) \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \\&\qquad +\,\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\sup _{\theta \in \Theta }\left\| \frac{\partial \tilde{m}_{\alpha ,t,2}(\theta )}{\partial \lambda _t(\theta ^*)}\left( \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}-\frac{\partial \tilde{\lambda }_t(\theta ^*)}{\partial \theta ^*}\right) \right\| \\&\quad \le \left( \left( \frac{\alpha ^2}{\delta _{L}^2}X_t^2+\left( \frac{\alpha }{\delta _{L}}+\frac{\alpha }{\delta _{L}^2}\right) X_t+\alpha ^2+\alpha \right) \sup _{\theta ^*\in \Theta ^*}\left\| \frac{\partial \lambda _t(\theta ^*)}{\partial \theta ^*}\right\| \right. \\&\qquad \left. +\frac{\alpha }{\delta _{L}}X_t+\alpha \right) V\rho ^t. \end{aligned}$$
\(\square \)
Using the fact that \(\bar{\lambda }_t(\theta ^*)\le \lambda _t(\theta ^*)+ |\lambda _t(\theta ^*)-\tilde{\lambda }_t(\theta ^*)|\) and Lemma 6 of Kim and Lee [18], the right hand side of the above inequality converges to 0 exponentially fast a.s., then the lemma is established. We can see Straumann and Mikosch [27] and Cui and Zheng [9] for details of the concept and properties of exponentially fast a.s. convergence.
Lemma A.7
Suppose that
$$\begin{aligned} \hat{\theta }_{\alpha ,n}^{H}=\arg \min _{\theta \in \Theta }H_{\alpha ,n}(\theta ), \end{aligned}$$
and conditions \((\mathrm{A}1){-}(\mathrm{A}8)\) hold. Then, we have
$$\begin{aligned} \hat{\theta }_{\alpha ,n}^{H}\xrightarrow {a.s.}\theta _0 \end{aligned}$$
and
$$\begin{aligned} \sqrt{n}(\hat{\theta }_{\alpha ,n}^{H}-\theta _0)\xrightarrow {d} N(0, J_\alpha ^{-1}K_{\alpha }{J_\alpha }^{-\top })~~as~~ n\rightarrow \infty . \end{aligned}$$
Proof
As in the proof of Theorem 3.4, we can see \(\sup _{\theta \in \Theta }\left| \frac{1}{n}\sum _{t=1}^{n}l_{\alpha ,t}(\theta )-E l_{\alpha ,t}(\theta )\right| \) converges to 0 a.s., and \(E l_{\alpha ,t}(\theta )\) has a unique minimum at \(\theta _0\) by Lemma lemma3A.3, then the first part of the lemma is validated. \(\square \)
Now, we verify the second part of the lemma. From Taylor’s expansion, we obtain
$$\begin{aligned} 0=\frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta } +\frac{1}{n}\sum _{t=1}^n\frac{\partial ^2 l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }} \sqrt{n}(\hat{\theta }_{\alpha ,n}^{H}-\theta _0), \end{aligned}$$
where \(\bar{\theta }_{\alpha ,n}\) is an intermediate point between \(\hat{\theta }_{\alpha ,n}^{H}\) and \(\theta _0\). First, we show that
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta }\xrightarrow {d} N(0, K_{\alpha }). \end{aligned}$$
(E1)
Note that for \(v={(v_1,v_2^{\top })}^{\top }\in \mathbb {R}\times \mathbb {R}^d\),
$$\begin{aligned} E\left( v^{\top }\frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta }\bigg |\mathcal {F}_{t-1}\right)= & {} v_1E\left( \frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \sigma }\bigg |\mathcal {F}_{t-1}\right) \\&+\,v_2^{\top }\frac{\partial \lambda _t(\theta _0^*)}{\partial \theta ^*}E\left( \frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \lambda _t(\theta ^*)}\bigg |\mathcal {F}_{t-1}\right) =0 \end{aligned}$$
and
$$\begin{aligned}E\left( v^{\top }\frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta }\right) ^2=v^{\top }E\left( \frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta }\frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta ^{\top }}\right) v<\infty , \end{aligned}$$
due to the second part of Lemma lemma5A.5A. Hence, using the central limit theorem in Billingsley [3], we obtain
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^nv^{\top }\frac{\partial l_{\alpha ,t}(\theta _0)}{\partial \theta }\xrightarrow {d} N(0, v^{\top }K_{\alpha }v), \end{aligned}$$
which asserts (E1). Now, we show that
$$\begin{aligned} -\frac{1}{n}\frac{\partial ^2l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}\xrightarrow {a.s} J_{\alpha }. \end{aligned}$$
(E2)
In view of the first part of Lemma lemma5A.5, \(J_{\alpha }\) is finite. Further, since
$$\begin{aligned}&v^{\top }(-J_{\alpha })v =(v_1~v_2^{\top }) \begin{pmatrix} &{}E\dfrac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \sigma ^2} &{}E\dfrac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \sigma \partial {\theta ^*}^{\top }}\\ &{}E\dfrac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \theta ^*\partial \sigma } &{}E\dfrac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \theta ^*\partial {\theta ^*}^{\top }}\\ \end{pmatrix} \left( {\begin{array}{c}v_1\\ v_2\end{array}}\right) \\&\quad =v_1\left( \frac{1+\alpha }{\sigma _0^4}\sum _{x=0}^{\infty } P_{\theta _0}^{1+\alpha }(x|\mathcal {F}_{t-1}) K_{\alpha ,t,1}^2(x;\theta _0)\right) v_1 -2v_2^{\top }\frac{\partial \lambda _t(\theta _0^*)}{\partial \theta ^*}\\&\qquad \left( \frac{1+\alpha }{\sigma _0^2}\sum _{x=0}^{\infty } P_{\theta _0}^{1+\alpha }(x|\mathcal {F}_{t-1}) K_{\alpha ,t,1}^2(x;\theta _0)\frac{1}{\sigma _0\lambda _t(\theta _0^*)+1}\left( \frac{x}{\lambda _t(\theta _0^*)}-1\right) \right) v_1\\&\qquad +v_2^{\top }\frac{\partial \lambda _t(\theta _0^*)}{\partial \theta ^*}\\&\qquad \left( (1+\alpha )\sum _{x=0}^{\infty } P_{\theta _0}^{1+\alpha }(x|\mathcal {F}_{t-1})\frac{1}{\sigma _0\lambda _t(\theta _0^*)+1}\left( \frac{x}{\lambda _t(\theta _0^*)}-1\right) ^2\right) \left( v_2^{\top }\frac{\partial \lambda _t(\theta _0^*)}{\partial \theta ^*}\right) ^{\top }\\&\quad >(1+\alpha )\sum _{x=0}^{\infty } P_{\theta _0}^{1+\alpha }(x|\mathcal {F}_{t-1})\\&\qquad \left( v_1\frac{1}{\sigma _0^2} K_{\alpha ,t,1}(x;\theta _0)-v_2^{\top }\left( \dfrac{\partial \lambda _t(\theta _0^*)}{\partial \theta ^*}\right) \frac{1}{\sigma _0\lambda _t(\theta _0^*)+1}\left( \frac{x}{\lambda _t(\theta _0^*)}-1\right) \right) ^{2}\ge 0, \end{aligned}$$
\(J_{\alpha }\) is non-singular with (A8). Note that
$$\begin{aligned}&\left| \left| \frac{1}{n}\sum _{t=1}^{n} \frac{\partial ^2 l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}- E\left( \frac{\partial ^2 l_t(\theta _0)}{\partial \theta \partial \theta ^{\top }}\right) \right| \right| \\&\quad \le \left| \left| \frac{1}{n}\sum _{t=1}^{n} \frac{\partial ^2 l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}- E\left( \frac{\partial ^2 l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}\right) \right| \right| \\&\qquad +\left| \left| E\left( \frac{\partial ^2 l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}\right) -E\left( \frac{\partial ^2 l_t(\theta _0)}{\partial \theta \partial \theta ^{\top }}\right) \right| \right| \\&\quad \le \sup _{\theta \in \Theta }\left| \left| \frac{1}{n}\sum _{t=1}^{n} \frac{\partial ^2 l_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^{\top }}- E\left( \frac{\partial ^2 l_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^{\top }}\right) \right| \right| \\&\qquad +\left| \left| E\left( \frac{\partial ^2 l_{\alpha ,t}(\bar{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}\right) -E\left( \frac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^{\top }}\right) \right| \right| . \end{aligned}$$
The stationarity and ergodicity of \(\partial ^2 l_{\alpha ,t}(\theta )/\partial \theta \partial \theta ^{\top }\) and the first part of Lemma lemma5A.5A imply that the first term in the last inequality converges to 0 a.s., moreover, the second term of the last inequality is no more than
$$\begin{aligned} E\left( \sup _{\Vert \theta -\theta _0\Vert \le \Vert \hat{\theta }_{\alpha ,n}^{H} -\theta _0\Vert } \left\| \frac{\partial ^2 l_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^{\top }}-\frac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^{\top }}\right\| \right) , \end{aligned}$$
(E3)
then by the dominated convergence theorem, we can see that (E3) decays to 0. So that (E2) is verified. Therefore, from (E1) and (E2), the second part of the lemma is established.
Proof of Theorem 3.5
From Taylor’s expansion, we have
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial l_{\alpha ,t}(\hat{\theta }_{\alpha ,n}^{H})}{\partial \theta }-\frac{1}{\sqrt{n}} \sum _{t=1}^n\frac{\partial l_{\alpha ,t}(\hat{\theta }_{\alpha ,n})}{\partial \theta } =\frac{1}{n}\sum _{t=1}^n\frac{\partial ^2 l_{\alpha ,t}(\tilde{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }} \sqrt{n}(\hat{\theta }_{\alpha ,n}^{H}-\hat{\theta }_{\alpha ,n}), \end{aligned}$$
where \(\tilde{\theta }_{\alpha ,n}\) is an intermediate point between \(\hat{\theta }_{\alpha ,n}^{H}\) and \(\hat{\theta }_{\alpha ,n}\). Furthermore, from the facts that \(n^{-1}\sum _{t=1}^{n}\partial l_{\alpha ,t}(\hat{\theta }_{\alpha ,n}^{H})/\partial \theta ^{\top }=0\) and \(n^{-1}\sum _{t=1}^{n}\partial \tilde{l}_{\alpha ,t}(\hat{\theta }_{\alpha ,n})/\partial \theta ^{\top }=0\), we obtain
$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial \tilde{l}_{\alpha ,t}(\hat{\theta }_{\alpha ,n})}{\partial \theta }-\frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial l_{\alpha ,t}(\hat{\theta }_{\alpha ,n})}{\partial \theta }\nonumber \\&\quad =\frac{1}{n}\sum _{t=1}^n\frac{\partial ^2 l_{\alpha ,t}(\tilde{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }} \sqrt{n}(\hat{\theta }_{\alpha ,n}^{H}-\hat{\theta }_{\alpha ,n}). \end{aligned}$$
The left hand side of the above equation converges to 0 a.s. by Lemma lemma6A.6A, and we can show that \(\frac{1}{n}\sum \nolimits _{t=1}^{n}\frac{\partial ^2 l_{\alpha ,t}(\tilde{\theta }_{\alpha ,n})}{\partial \theta \partial \theta ^{\top }}\) converges to \(E\left( \frac{\partial ^2 l_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^{\top }}\right) \) a.s. in a similar manner as in the proof of Lemma lemma7A.7A. Therefore, the theorem is established by Lemma lemma7A.7A. \(\square \)
