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)= \frac12{ d^{ 2}\over dx^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 $G'$ is a generalized derivative of $G$ and $:(\cd)^{ k}:$ is the $k$--th order Wick power.
Categories: math.PR