Asymptotic expansions for functions of the increments of certain Gaussian processes

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 $Ef^{2}(η)<\ff$, where $\eta =N(0,1)$. When ${\sigma }^2$ is regularly varying at zero and \[ \lim_{h\to 0}{h^2\over σ^2(h)}= 0\qquad {and}\qquad \lim_{h\to 0}{σ^2(h)\over h}= 0 \quad {but} \quad ({d^{2}\over ds^2}σ^2(s))^{j_0} \] 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)}{σ(h)}) dx \label{abst}\nn &&\qquad = \sum_{j=0}^{j_0} (h/σ(h))^{j} {E(H_{j}(η) f(η))\over\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\overσ(h)})^{j_0}\nn \eea in $L^2$. Here $H_j$ is the $j$-th Hermite polynomial. Also $:(G^{\prime }{)}^j:(I_{[a,b]})$ is a $j$-th order Wick power Gaussian chaos constructed from the Gaussian field $G^{\prime }(g)$, with covariance \[ E(G'(g)G'(\wt g)) = \int \int ρ(x-y)g(x)\wt g(y) dx dy\label{3.7bqs}, \] where $\rho (s)=1/2\frac{d^2}{d{s}^2}{\sigma }^2(s)$.

References in corpus (1)

• Noncentral convergence of multiple integrals