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

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