Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

Abstract

We study the asymptotic behavior of the $\nu$-symmetric Riemann sums for functionals of a self-similar centered Gaussian process X with increment exponent $0<\alpha <1$. We prove that, under mild assumptions on the covariance of X, the law of the weak $\nu$-symmetric Riemann sums converge in the Skorohod topology when $\alpha =(2\ell +1)^{-1}$, where $\ell$ denotes the largest positive integer satisfying $\int _{0}^{1}x^{2j}\nu (\mathrm{d}x)=(2j+1)^{-1}$ for all $j=0,\dots , \ell -1$. In the case $\alpha >(2\ell +1)^{-1}$, we prove that the convergence holds in probability.

Appendix

The following lemmas are estimations on the covariances of increments of X. The proof of these results relies on some technical lemmas proved by Nualart and Harnett in [12]. In what follows C is a generic constant depending only on the covariance of the process X.

Lemma 5.1

Under (H.1), for $0<s\le t$ we have

$$\begin{aligned} {\mathbb E}\left[ (X_{t+s} - X_t)^2\right] = 2\lambda t^{2\beta -\alpha }s^\alpha + g_1(t,s), \end{aligned}$$

where $|g_1(t,s)| \le C st^{2\beta -1}$.

Proof

See [12, Lemma 3.1] and notice that the proof only uses that $|\psi '|$ is bounded in (1, 2]. $\square$

Remark 5.2

Notice that $g_1(t,s)$ satisfies $|g_1(t,s)| \le C s^{\alpha } t^{2\beta -\alpha }$, because $\alpha <1$ and $\alpha \le 2\beta$. Therefore, for any $0<s\le t$, we obtain

$$\begin{aligned} {\mathbb E}\left[ (X_{t+s} - X_t)^2\right] \le C s^{\alpha } t^{2\beta -\alpha }. \end{aligned}$$

With the notation of Sect. 2.3, this implies

$$\begin{aligned} \xi ^2_{j,n} \le Cn^{-2\beta } j^{2\beta -\alpha }. \end{aligned}$$

(5.1)

On the other hand, we deduce that for every $T>0$, there exists $C>0$, which depends on T and the covariance of X, such that

$$\begin{aligned} \sup _{0\le t\le \left\lfloor nT\right\rfloor }\mathbb {E}\left[ \Delta X_{\frac{t}{n}}^{2}\right] \le C n^{-\alpha }. \end{aligned}$$

(5.2)

Lemma 5.3

Let j, k, n be integers with $n\ge 6$ and $1\le j\le k$. Under (H.1)-(H.2), we have the following estimates:

• (a) If $j+3\le k\le 2j+2$, then

  $$\begin{aligned} \left| \mathbb {E}\left[ \Delta X_{\frac{j}{n}}\Delta X_{\frac{k}{n}} \right] \right| \le Cn^{-2\beta }j^{2\beta -\alpha }k^{\alpha -2}. \end{aligned}$$

  (5.3)

• (b) If $k\ge 2j+2$, then

  $$\begin{aligned} \left| \mathbb {E}\left[ \Delta X_{\frac{j}{n}}\Delta X_{\frac{k}{n}} \right] \right| \le Cn^{-2\beta }j^{2\beta +\nu -2}k^{-\nu }. \end{aligned}$$

  (5.4)

Proof

We have

$$\begin{aligned} {\mathbb E}\left[ \Delta X_{\frac{k}{n}} \Delta X_{\frac{j}{n}}\right]&= n^{-2\beta }(j+1)^{2\beta }\left( \phi \left( \frac{k+1}{j+1}\right) - \phi \left( \frac{k}{j+1}\right) \right) \\&\qquad \quad -\,n^{-2\beta }j^{2\beta }\left( \phi \left( \frac{k+1}{j}\right) - \phi \left( \frac{k}{j}\right) \right) \\&= n^{-2\beta }\left( (j+1)^{2\beta }-j^{2\beta }\right) \left( \phi \left( \frac{k+1}{j+1}\right) - \phi \left( \frac{k}{j+1}\right) \right) \\&\qquad \quad +\,n^{-2\beta }j^{2\beta }\left[ \phi \left( \frac{k+1}{j+1}\right) - \phi \left( \frac{k}{j+1}\right) - \phi \left( \frac{k+1}{j}\right) + \phi \left( \frac{k}{j}\right) \right] . \end{aligned}$$

We first show (5.3). Condition $j+3 \le k \le 2j+2$ implies that the interval $\left[ \frac{k}{j+1}, \frac{k+1}{j} \right]$ is included in the interval [1, 5]. Therefore, using (1.8) and (1.9), we deduce that there exists a constant $C>0$ such that for all $x\in \left[ \frac{k}{j+1}, \frac{k+1}{j} \right]$,

$$\begin{aligned} \left| \phi ' \left( x \right) \right| \le C(k/j)^{\alpha -1}. \end{aligned}$$

and

$$\begin{aligned} \left| \phi ''\left( x \right) \right| \le C(k/j)^{\alpha -2}. \end{aligned}$$

The estimate (5.3) follows easily from the Mean Value Theorem.

On the other hand, $k\ge 2j+2$ implies that the interval $\left[ \frac{k}{j+1}, \frac{k+1}{j} \right]$ is included in the interval $[2, \infty ]$. Therefore, using (1.8) and (1.9), we deduce that there exists a constant $C>0$ such that for all $x\in \left[ \frac{k}{j+1}, \frac{k+1}{j} \right]$,

$$\begin{aligned} \left| \phi ' \left( x \right) \right| \le C(k/j)^{-\nu }. \end{aligned}$$

and

$$\begin{aligned} \left| \phi ''\left( x \right) \right| \le C(k/j)^{-\nu -1}. \end{aligned}$$

Therefore, estimate (5.4) follows easily from the Mean Value Theorem. The proof of the lemma is now complete. $\square$

Last, we have two technical results that have been used in the proofs of Theorems 1.5 and 1.6. For a fixed integer n and nonnegative real $t_{1},t_{2}$, note that the notation of Sect. 2.3 gives

$$\begin{aligned} \mathbb {E}[\Delta X_{\frac{t_{1}}{n}}\Delta X_{\frac{t_{2}}{n}}]=\left\langle \partial _{\frac{t_{1}}{n}},\partial _{\frac{t_{2}}{n}} \right\rangle _{{\mathfrak H}}. \end{aligned}$$

Lemma 5.4

Assume X satisfies (H.1) and (H.2). Then, for any integer $n\ge 2$ and real $T>0$, there is a constant C is a constant which depends on T and the covariance of X, such that

$$\begin{aligned} \sup _{0\le k\le \lfloor nT \rfloor -1}\sum _{j=0}^{\lfloor nT \rfloor -1}\left| \left\langle \partial _{\frac{j}{n}}, \partial _{\frac{k}{n}}\right\rangle _{\mathfrak H}\right| \le Cn^{-\alpha }. \end{aligned}$$

(5.5)

Proof

In view of the estimate (5.2), we can assume that $n\ge 6$ and $4\le j+3 \le k$ or $4\le k+3 \le j$. If $4\le j+3 \le k$, from the estimates (5.3) and (5.4), we deduce

$$\begin{aligned} \left| \left\langle \partial _{\frac{j}{n}}, \partial _{\frac{k}{n}}\right\rangle _{\mathfrak H}\right| \le C n^{-2\beta } j^{2\beta -2}. \end{aligned}$$

Summing in the index j, we get the desired result, because $2\beta -1 \le 0$ and $2\beta \ge \alpha$. On the other hand, if $4\le k+3 \le j\le 2k+2$, the estimates (5.3) yields

$$\begin{aligned} \left| \left\langle \partial _{\frac{j}{n}}, \partial _{\frac{k}{n}}\right\rangle _{\mathfrak H}\right| \le C n^{-2\beta } k^{2\beta -\alpha } j^{\alpha -2} \le C n^{-\alpha } j^{\alpha -2}, \end{aligned}$$

which gives the desired estimate. Finally, if $4\le k+3$ and $2k+2 \le j$, the estimate (5.4) yields

$$\begin{aligned} \left| \left\langle \partial _{\frac{j}{n}}, \partial _{\frac{k}{n}}\right\rangle _{\mathfrak H}\right| \le C n^{-2\beta } k^{2\beta +\nu -2} j^{-\nu }. \end{aligned}$$

If $\alpha +\nu -2 \le 0$, then summing the above estimate in j we obtain the bound

$$\begin{aligned} Cn^{-2\beta } k^{2\beta -\alpha + (\alpha +\nu -2)} \le Cn^{-\alpha }. \end{aligned}$$

On the other hand, if $\alpha +\nu -2 >0$, then

$$\begin{aligned} C n^{-2\beta } k^{2\beta +\nu -2} j^{-\nu } \le C n^{-2\beta } k^{2\beta -\alpha } \left( \frac{k}{j} \right) ^{\alpha +\nu -2} j^{\alpha -2} \le C n^{-\alpha } j^{\alpha -2} \end{aligned}$$

and summing in j we get the desired bound. $\square$

Lemma 5.5

Assume that $0<\alpha <1$ and let $n\ge 1$ be an integer. Then, for every $r\in \mathbb {N}$ and $T\ge 0$,

$$\begin{aligned} \sum _{j=0}^{\left\lfloor nT\right\rfloor -1}\left| \left\langle \partial _{\frac{j}{n}},\widetilde{\varepsilon }_{\frac{j}{n}} \right\rangle _{{\mathfrak H}}\right| ^{r}\le Cn^{-2\beta (r-1)}. \end{aligned}$$

(5.6)

Proof

By (1.2),

$$\begin{aligned} \left\langle \partial _{\frac{j}{n}},\widetilde{\varepsilon }_{\frac{j}{n}} \right\rangle _{{\mathfrak H}} =\frac{1}{2}\mathbb {E}\left[ (X_{\frac{j+1}{n}}-X_{\frac{j}{n}})(X_{\frac{j+1}{n}}+X_{\frac{j}{n}})\right] =\frac{1}{2}\mathbb {E}\left[ X_{\frac{j+1}{n}}^2-X_{\frac{j}{n}}^2\right] =\phi (1)\Psi _{n}(j), \end{aligned}$$

where

$$\begin{aligned} \Psi _{n}(j) =\left( \left( \frac{j+1}{n}\right) ^{2\beta }-\left( \frac{j}{n}\right) ^{2\beta }\right) . \end{aligned}$$

We can easily show that $\Psi _{n}(j)\le C n^{-2\beta }$, and hence,

$$\begin{aligned} \sum _{j=0}^{\left\lfloor nT\right\rfloor -1}\left| \left\langle \partial _{\frac{j}{n}},\widetilde{\varepsilon }_{\frac{j}{n}} \right\rangle _{{\mathfrak H}}\right| ^r =\phi (1)^{r}\sum _{j=0}^{\left\lfloor nT\right\rfloor -1}\Psi _{n}(j)^r\le Cn^{-2\beta (r-1)}\sum _{j=0}^{\left\lfloor nT\right\rfloor -1}\Psi _{n}(j). \end{aligned}$$

Since the right-hand side of the last inequality is a telescopic sum, we get

$$\begin{aligned} \sum _{j=0}^{\left\lfloor nT\right\rfloor -1}\left| \left\langle \partial _{\frac{j}{n}},\widetilde{\varepsilon }_{\frac{j}{n}} \right\rangle _{{\mathfrak H}}\right| ^r \le Cn^{-2\beta (r-1)}\left( \frac{\left\lfloor nT\right\rfloor }{n}\right) ^{2\beta }. \end{aligned}$$

Relation (5.6) follows from the previous inequality. $\square$