Asymptotic expansions for functions of the increments of certain Gaussian processes

TL;DR

This paper derives asymptotic expansions for functions of increments of Gaussian processes with stationary increments, revealing their structure in terms of Hermite polynomials and Gaussian chaos under specific regularity conditions.

Contribution

It provides a novel asymptotic expansion for integrals of functions of Gaussian process increments, involving Hermite polynomials and Wick powers, under new regularity conditions.

Findings

Asymptotic expansion in $L^2$ for functions of Gaussian increments

Representation involving Hermite polynomials and Gaussian chaos

Conditions on $\sigma^2$ for the expansion to hold

Abstract

Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $\sigma^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(\eta)<\ff$, where $\eta=N(0,1)$. When $\sigma^2$ is regularly varying at zero and \[ \lim_{h\to 0}{h^2\over \sigma^2(h)}= 0\qquad {and}\qquad \lim_{h\to 0}{\sigma^2(h)\over h}= 0 \quad {but} \quad ({d^{2}\over ds^2}\sigma^2(s))^{j_0} \] is locally integrable for some integer $j_0\ge 1$, and satisfies some additional regularity conditions, \bea && \int_a^bf(\frac{G(x+h)-G(x)}{\sigma (h)}) dx \label{abst}\nn &&\qquad = \sum_{j=0}^{j_0} (h/\sigma(h))^{j} {E(H_{j}(\eta) f(\eta))\over\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\over\sigma (h)})^{j_0}\nn \eea in $L^2$. Here $H_j$ is the $j$-th Hermite polynomial. Also $:(G')^{j}:(I_{[a,b]})$ is a $j $-th order Wick power Gaussian chaos constructed from the Gaussian field $ G'(g) $, with covariance \[ E(G'(g)G'(\wt g)) = \int \int \rho (x-y)g(x)\wt g(y) dx dy\label{3.7bqs}, \] where $ \rho(s)={1/2}{d^{2}\over ds^2}\sigma^2(s)$.