An almost sure limit theorem for Wick powers of Gaussian differences quotients

Abstract

Let G={G(x), x\in R_+}, G(0)=0, be a mean zero Gaussian process with $E(G(x)-G(y){)}^2={\sigma }^2(x-y)$. Let $\rho (x)=\frac{1}{2}\frac{d^2}{d{x}^2}{\sigma }^2(x)$, $x\ne 0$. When ${\rho }^k$ is integrable at zero and satisfies some additional regularity conditions, \[ \lim_{h\downarrow 0} \int :(\frac{G(x+h)-G(x)}{h})^{k}:g(x) dx=:(G') ^{k}:(g){.3 in}a.s. \] for all $g\in B_0(R^{+})$, the set of bounded Lebesgue measurable functions on $R_{+}$ with compact support. Here $G^{\prime }$ is a generalized derivative of $G$ and $:(\cd)^{k}:$ is the $k$--th order Wick power.