Local times and sample path properties of the Rosenblatt process

Abstract

Let $Z={\left(Z_t\right)}_{t\ge 0}$ be the Rosenblatt process with Hurst index $H\in (1∕2,1)$. We prove joint continuity for the local time of $Z$, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

Introduction

The Rosenblatt process $Z={\left(Z_t\right)}_{t\ge 0}$ is a self-similar stochastic process with long-range dependence and heavier tails than those of the normal distribution. It depends on a parameter $H\in (\frac{1}{2},1)$ which is fixed in what follows. The process $Z$ belongs to the family of Hermite processes that naturally arise as limits of normalized sums of long-range dependent random variables [8]. The first Hermite process is the fractional Brownian motion, which is Gaussian and thus satisfies many desirable properties. The Rosenblatt process is the second Hermite process: it is no longer Gaussian and was introduced by Rosenblatt in [20]. Although less popular than fractional Brownian motion, the Rosenblatt process is attracting increasing interest in the literature, mainly due to its self-similarity, stationarity of increments, and long-range dependence properties. See for example [1], [6], [15], [18] for recent studies on different aspects of this process.

The Rosenblatt process ${\left(Z_t\right)}_{t\ge 0}$ admits the following stochastic representation, also known as the spectral representation:

$$Z_t={\int }_{{\mathbb{R}}^2}{H}_t(x,y)Z_G\left(dx\right)Z_G\left(dy\right).$$

In (1.1), the double Wiener–Itô integral is taken over $x\ne \pm y$ and $Z_G\left(dx\right)$ is a complex-valued random white noise with control measure $G$ satisfying $G\left(dx\right)={\left|x\right|}^{-H}dx$. The kernel $H_t(x,y)$ is given by

$$H_t(x,y)=\frac{e^{it(x+y)}-1}{i(x+y)},$$

and is a complex valued Hilbert–Schmidt kernel satisfying $H_t(x,y)=H_t(y,x)=\overline{H_t(-x,-y)}$ and ${\int }_{{\mathbb{R}}^2}{\left|H_t(x,y)\right|}^{2}G\left(dx\right)G\left(dy\right)<\infty$. In particular, the spectral theorem applies, see [8], and allows one to write

$$Z_t\stackrel{d}{=}\sum _{k=1}^{\infty }{\lambda }_k(X_k^2-1),$$

where ${\left(X_k\right)}_{k\ge 1}$ is a sequence of independent standard Gaussian random variables and ${\left({\lambda }_k\right)}_{k\ge 1}$ are the eigenvalues¹ (repeated according to the possible multiplicity) of the self-adjoint operator $A:L_G^2\left(\mathbb{R}\right)\to L_G^2\left(\mathbb{R}\right)$, given by

$$\left(Af\right)\left(x\right)≔{\int }_{\mathbb{R}}{H}_t(x,-y)f\left(y\right)G\left(dy\right)={\int }_{\mathbb{R}}{H}_t(x,-y)f\left(y\right){\left|y\right|}^{-H}dy.$$

Furthermore, ${\sum }_{k\ge 1}{\lambda }_k^2<\infty$ and thus (1.3) converges.

The goal of the present paper is to study the occupation density (also known as the local time) $L(x,I)$ of $Z$, where $x\in \mathbb{R}$ and $I\subset (0,\infty )$ is a finite closed interval. Recall that for a deterministic function $f:{\mathbb{R}}_{+}\to \mathbb{R}$, the occupation measure of $f$ is defined by

$$\nu (A,B)=\mu (B\cap f^{-1}\left(A\right)),$$

where $A\subset \mathbb{R}$ and $B\subset {\mathbb{R}}_{+}$ are Borel sets and $\mu$ is the Lebesgue measure on ${\mathbb{R}}_{+}$. Observe that $\nu (A,B)$ represents the amount of time during a period $B$ where $f$ takes value in $A$. Then, when $\nu (\cdot ,B)$ is absolutely continuous with respect to $\mu$, the occupation density (or local time) is given by the Radon–Nikodym derivative

$$L(x,B)=\frac{d\nu }{d\mu }(x,B).$$

We study the local time via the analytic approach initiated by Berman [4]. The idea is to relate properties of $L(x,B)$ with the integrability properties of the Fourier transform of $f$. Recall the following key result [10]:

Proposition 1.1

The function $f$ has an occupation density $L(x,B)$ for $x\in \mathbb{R},B\in \mathcal{B}\left([u,U]\right)$ which is square integrable in $x$ for every fixed $B$ iff

$${\int }_{\mathbb{R}}{|{\int }_u^{U}exp\left(i\xi f\left(t\right)\right)dt|}^{2}d\xi <\infty .$$

Moreover, in this case, the occupation density can be represented as

$$L(x,B)=\frac{1}{2\pi }{\int }_{\mathbb{R}}{\int }_{B}exp\left(i\xi (x-f\left(s\right))\right)d\xi ds.$$

The deterministic function $f\left(t\right)$ can be chosen to be the single path of a stochastic process ${\left(X_t\right)}_{t\ge 0}$. To show almost sure existence and square integrability of $L(x,B)$ in that case, it will be enough to show that

$$\mathbb{E}\left[{\int }_{\mathbb{R}}{|{\int }_u^{U}exp\left(i\xi X_t\right)dt|}^{2}d\xi \right]<\infty ,$$

or equivalently

$${\int }_{\mathbb{R}}{\int }_u^U{\int }_u^U\mathbb{E}[exp\left(i\xi (X_s-X_t)\right)]dsdtd\xi <\infty .$$

If ${\left(X_t\right)}_{t\ge 0}$ is Gaussian, then one can evaluate $\mathbb{E}[exp\left(i\xi (X_s-X_t)\right)]$ explicitly to establish (1.4). It leads to the well-known Gaussian criterion:

Proposition 1.2 Lemma 19 in [10]

Suppose that $X$ is Gaussian and centered, and satisfies

$${\int }_{{[u,U]}^2}\Delta {(s,t)}^{-1∕2}dsdt<\infty ,$$

where $\Delta (s,t)=\mathbb{E}\left[{(X_s-X_t)}^2\right]$. Then ${\left(X_t\right)}_{t\ge 0}$ has an occupation density $L=L(x,B,\omega )$ which, for any $B\in \mathcal{B}\left([u,U]\right)$, is square integrable, in particular (1.4) holds.

In our setting ${\left(X_t\right)}_{t\ge 0}={\left(Z_t\right)}_{t\ge 0}$ is the Rosenblatt process which is not Gaussian. Nevertheless, a careful analysis of $\mathbb{E}[exp\left(i\xi (Z_s-Z_t)\right)]$ via (1.3) yields (1.4). This is the approach in [21] where existence of the local time of the Rosenblatt process was first established. In this paper we show a considerably more involved bound on the Fourier transform which in turn will yield deeper results regarding the irregularity properties of the sample paths. The following is the key result of our paper:

Proposition 1.3

Let $n\in \mathbb{N}$ and $0\le \eta <\frac{1-H}{2H}$. Then, for any times $0\le u<U$, the Rosenblatt process satisfies

$${\int }_{{[u,U]}^n}{\int }_{{\mathbb{R}}^n}\prod _{j=1}^n{\left|{\xi }_j\right|}^{\eta }|\mathbb{E}exp\left(i\sum _{j=1}^n{\xi }_j{Z}_{t_j}\right)|d\xi dt\le C^n{n}^{2nH(1+\eta )}{(U-u)}^{(1-H(1+\eta ))n},$$

where the constant $C>0$ depends only on $H$ and $\eta$.

Proposition 1.3 can be applied to obtain the following Hölder conditions on $L(x,B)$.

Theorem 1.4

Let ${\left(Z_t\right)}_{t\ge 0}$ be a Rosenblatt process with $H\in \left(\frac{1}{2},1\right)$. The local time $(x,t)\mapsto L(x,[0,t])$ is almost surely jointly continuous and has finite moments. For a finite closed interval $I\subset (0,\infty )$, let $L^{\ast }\left(I\right)={sup}_{x\in \mathbb{R}}L(x,I)$. There exist deterministic constants $C_1$ and $C_2$ such that, almost surely,

$$\underset{r\to 0}{lim\; sup}\frac{L^{\ast }\left([s-r,s+r]\right)}{r^{1-H}{(loglog{r}^{-1})}^{2H}}\le C_1,$$

for any $s\in I$ and

$$\underset{r\to 0}{lim\; sup}\underset{s\in I}{sup}\frac{L^{\ast }\left([s-r,s+r]\right)}{r^{1-H}{(log{r}^{-1})}^{2H}}\le C_2.$$

In particular, the local time $L(x,I)$ is well defined for any fixed $x$ and interval $I\subset (0,\infty )$. Explicit estimates for the moments of the local time are provided in Theorem 3.1. As a direct corollary we obtain:

Corollary 1.5

For any finite closed interval $I\subset (0,\infty )$ there exist deterministic constants $C_1$ and $C_2$, independent of $x$ and $t$, such that, almost surely, for every $t\in I$ and every $x\in \mathbb{R}$

$$\underset{r\to 0}{lim\; sup}\frac{L(x,[t-r,t+r])}{r^{1-H}{(loglog{r}^{-1})}^{2H}}\le C_1,$$

and for every $x\in \mathbb{R}$

$$\underset{r\to 0}{lim\; sup}\underset{t\in I}{sup}\frac{L(x,[t-r,t+r])}{r^{1-H}{(log{r}^{-1})}^{2H}}\le C_2.$$

Moreover, we get the following for a particular Hausdorff measure:

Corollary 1.6

Let $I\subset (0,\infty )$ be a finite closed interval. There exists a deterministic constant $C$ such that for every $x\in \mathbb{R}$ we have

$${\mathcal{H}}_{\varphi }(Z^{-1}\left(x\right)\cap I)\ge CL(x,I),\text{ a.s., }$$

where ${\mathcal{H}}_{\varphi }$ denotes $\varphi$-Hausdorff measure with $\varphi \left(r\right)=r^{1-H}{(loglog{r}^{-1})}^{2H}$.

Furthermore, we can quantify the behavior of the trajectories of $Z$. We also point the reader to the recent paper [2] for a similar result obtained with a different approach.

Corollary 1.7

Let $I\subset (0,\infty )$ be a finite closed interval. There exists a deterministic constant $C>0$ such that for every $s\in I$ we have, almost surely,

$$\underset{r\to 0}{lim\; inf}\underset{s-r<t<s+r}{sup}\frac{|Z_t-Z_s|}{r^H{(loglog{r}^{-1})}^{-2H}}\ge C,$$

and

$$\underset{r\to 0}{lim\; inf}\underset{s\in I}{inf}\underset{s-r<t<s+r}{sup}\frac{|Z_t-Z_s|}{r^H{(log{r}^{-1})}^{-2H}}\ge C.$$

In particular, $Z$ is almost surely nowhere differentiable.

In Section 2 we establish our main result Proposition 1.3. As we are dealing with a second order Hermite process, we are forced to control from below singular values of somewhat unwieldy operators (see Remark 5.3). For this purpose we need to introduce several technical lemmas exhibiting tools from operator theory and harmonic analysis, including the theory of weighted integrals. Their proofs are postponed into Section 5. Section 3 is dedicated to some results regarding the existence and joint continuity of the local time. In particular, bounds on the moments of $L(x,B)$ are obtained. Finally, in Section 4, Theorem 1.4 and Corollary 1.5, Corollary 1.6, Corollary 1.7 are established.