Asymptotic expansions for functions of the increments of certain Gaussian processes
Let $G=\{G(x),x\ge 0\}$ be a mean zero Gaussian process with stationary increments and set $σ^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(η)<\ff$, where $η=N(0,1)$. When $σ^2$ 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 $:(G')^{j}:(I_{[a,b]})$ is a $j $-th order Wick power Gaussian chaos constructed from the Gaussian field $ G'(g) $, with covariance \[ E(G'(g)G'(\wt g)) = \int \int ρ(x-y)g(x)\wt g(y) dx dy\label{3.7bqs}, \] where $ ρ(s)={1/2}{d^{2}\over ds^2}σ^2(s)$.