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

Abstract

Let G={G(x), x∈R₊}, G(0)=0, be a mean zero Gaussian process with E(G(x)−G(y))²=σ²(x−y). Let ρ(x)=½ d²/dx² σ²(x), x≠0. When ρ^k is integrable at zero and satisfies some additional regularity conditions,

lim_h↓0∫ : ((G(x+h)−G(x))/h)^k : g(x) dx= : (G')^k : (g) a.s.

for all g∈$\mathcal{B}$₀(R⁺), the set of bounded Lebesgue measurable functions on R₊ with compact support. Here G' is a generalized derivative of G and : ( ⋅ )^k : is the k–th order Wick power.