  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)$.

Probability