Modern Stochastics: Theory and Applications

Let $H>\frac{1}{2}$ and $\lambda >0$. The process where $(x)_{+}=xI(x>0)$ and $d=\frac{1}{2}-\frac{1-H}{k}\in (\frac{1}{2}-\frac{1}{2k},\infty )$, is called a tempered Hermite process of order k.

The function is well defined in ${L}^{2}({\mathbb{R}}^{k})$ for any $H>\frac{1}{2}$ and $\lambda >0$.

To show that $h_{t}(y_{1},\dots ,y_{k})$ is in ${L}^{2}({\mathbb{R}}^{k})$, we write where we applied the standard integral formula [13, p. 344] for $|\arg \beta |<\pi $, Re $\mu >0$, Re $\nu >0$. Here $K_{\nu }(x)$ is the modified Bessel function of the second kind (see [1, Chapter 9]). Next, we need to show that is finite for $d>\frac{1}{2}-\frac{1}{2k}$ (equivalently, for $H>\frac{1}{2}$). First, suppose $\frac{1}{2}-\frac{1}{2k}<d<\frac{1}{2}$ (or $\frac{1}{2}<H<1$). Since $k_{\nu }(z)<{z}^{-\nu }{2}^{\nu -1}\varGamma (\nu )$ for $z>0$ (Theorem 3.1 in [11]), we have which is finite, and, consequently, from (4) and (6) we get for $\frac{1}{2}-\frac{1}{2k}<d<\frac{1}{2}$. Next, suppose $d>\frac{1}{2}$ (equivalently $H>1$). In this case, where we applied the fact that $K_{\nu }(z)=K_{-\nu }(z)$ and $k_{\nu }(z)<{z}^{-\nu }{2}^{\nu -1}\varGamma (\nu )$ for $z>0$. Hence, from (4) and (7) it follows that for $d>\frac{1}{2}$. Finally, let $d=\frac{1}{2}$ (equivalently, $H=1$). Consider Since $K_{0}(z)\sim -\log (z)$ as $z\to 0$ (see [1, Eq. 9.6.8], we have Now, we find an upper bound for $I_{2}$. It can be shown that $\frac{K_{\nu }(x)}{K_{\nu }(y)}>{e}^{y-x}$ for $0<x<y$ and an arbitrary real number ν (see [4]). Therefore, From (8), (9), and (10) we can see that and hence for $\epsilon >0$, and this shows that for $d=\frac{1}{2}$ ($H=1$), which completes the proof.  □

The process ${Z_{H,\lambda }^{k}}$ given by (2) has stationary increments such that for any scale factor $c>0$.

Since $B(dy)$ has the control measure $m(dy)={\sigma }^{2}\hspace{0.1667em}dy$, the random measure $B(c\hspace{0.1667em}dy)$ has the control measure ${c}^{1/2}{\sigma }^{2}\hspace{0.1667em}dy$. Given $t_{j}$, $j=1,\dots ,n$, by the change of variables $s=c{s^{\prime }}$ and $y_{i}=c{y^{\prime }_{i}}$ for $i=1\dots ,k$ we have so that (12) holds. Suppose now that $s_{j}<t_{j}$ and change the variables $x={x^{\prime }}+s_{j}$, $y_{i}=s_{j}+{y^{\prime }_{i}}$ (for $j=1,\dots ,n$) to get which shows that a tempered Hermite process of order k has stationary increments.  □

The stochastic process ${Z_{H,\lambda }^{k}}(t)$ has a continuous version.

According to the proof of Lemma 1, where $c_{1}$ and $c_{2}$ are some constants. Kolmogorov’s continuity criterion states that a stochastic process $X(t)$ has a continuous version if there exist some positive constants p, β, and c such that Apply (14) for tempered Hermite process ${Z_{H,\lambda }^{k}}(t)$ by taking $p=1$, $\beta =\min \{1,2H-1\}$, and $c=\min \{c_{1},c_{2}\}$ to get the desired result.  □

The process ${Z_{H,\lambda }^{k}}$ given by (2) has the covariance function for $\lambda >0$ and $d>\frac{1}{2}-\frac{1}{2k}$ (equivalently, $H>\frac{1}{2}$).

By applying the Fubini theorem and the isometry of multiple Wiener–Itô integrals we have for any $H>\frac{1}{2}$ and $\lambda >0$, and hence we get the desired result.  □

Let for $N=1,2,\dots $, where $g_{N}\in {L}^{2}({\mathbb{Z}}^{k})$ for $k\ge 1$, and $\{\varepsilon _{n}\}$ is an i.i.d. sequence with mean zero and variance 1. Assume that, for some $f\in {L}^{2}({\mathbb{R}}^{k})$, where Then as $N\to \infty $.

See Proposition 4.1 in [3] and also Corollary 4.7.1 in [12].  □

Let ${Y}^{\lambda ,k}(n)$ be the discrete chaos process given by (18). Then where ⇒ means weak convergence in the Skorokhod space $D[0,1]$ with uniform metric, ${Z_{H,\lambda }^{k}}(t)$ is the tempered Hermite process in (2), and $H=1+kd-\frac{k}{2}$.

The Lamperti’s theorem [15] states that if and $d(N)\to \infty $ as $N\to \infty $, where $\{Y_{k}\}$ is stationary, then $\{Z(t)\}_{t\ge 0}$ is self-similar with stationary increments ($\stackrel{f.d.d.}{\longrightarrow }$ means the convergence of finite-dimensional distributions). In our case, since the stationary processes $\{{Y_{k}^{\frac{\lambda }{N}}}\}$ depend on N through the parameter λ, the limit process $\{{Z_{H,\lambda }^{k}}(t)\}_{t\ge 0}$ need not be a self-similar process. Therefore, the result of Theorem 1 does not contradict the Lamperti theorem.

First, we show that where and $Q_{k}(\cdot )$ is defined by (17). In order to show (21), we just need to check that as $N\to \infty $, where and $h_{t}(y_{1},\dots ,y_{k})$ is given by (3). Write Let $d=1$. In this case, for all $N\ge 1$, and hence Next, consider $0<d<1$. Since $[Ns]-[Ny]>Ns-Ny-1$, we get for all $N\ge 1$, and hence Finally, suppose that $d>1$. Since $[Ns]-[Ny]<Ns-Ny+1$, we get for all $N\ge 1$, and hence By the similar argument of Lemma 1, we can verify that for $i=1,2,3$. On the other hand, since ${C}^{\lambda }(i_{1},\dots ,i_{k})$ is continuous a.e., ${C}^{\lambda }(\frac{[Ns]-[Ny_{1}]}{N},\dots ,\frac{[Ns]-[Ny_{k}]}{N})$ converges a.e. to ${C}^{\lambda }(s-y_{1},\dots ,s-y_{k})$ as $N\to \infty $. Now apply the dominated convergence theorem to get the desired result (22).

In order to show the tightness, we need to verify that where $\gamma >0$ and $\alpha >\frac{1}{2}$ (here $\{F_{n}\}_{n\ge 1}$ is a sequence of nondecreasing continuous functions on $[0,1]$ that are uniformly bounded and satisfy where $\omega _{\delta }(F):=\sup _{|t-s|<\delta }|F(t)-F(s)|$ for $\delta >0$). See Lemma 4.4.1 in [12] for more details. Observe that Therefore, we get Now, we consider two different cases corresponding with the range of d. First, assume that $\frac{1}{2}-\frac{1}{2k}<d\le 1$ (equivalently, $\frac{1}{2}<H\le 1+\frac{k}{2}$): Since $[Ny]-[Nx_{j}]+1>Ny-Nx_{j}$, we can write Now, let $d>1$. Since $Ny-Nx_{j}<[Ny]-[Nx_{j}]+1<Ny-Nx_{j}+N$, we have Therefore, for any $d>\frac{1}{2}-\frac{1}{2k}$ (equivalently, $H>\frac{1}{2}$). According to the proof of Lemma 1, for $\frac{1}{2}<H<1$ and for $H>1$. Now, it remains to apply (23) by selecting $\gamma =1$, $\alpha =\min \{H,1\}$, and $F_{n}(t)=t$ to get the desired result.  □