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VOL. 5 | 2009 An almost sure limit theorem for Wick powers of Gaussian differences quotients
Michael B. Marcus, Jay Rosen

## Abstract

Let G={G(x), xR+}, G(0)=0, be a mean zero Gaussian process with E(G(x)−G(y))2=σ2(xy). Let ρ(x)=½ d2/dx2σ2(x), x≠0. When ρk is integrable at zero and satisfies some additional regularity conditions,

limh↓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.

## Information

Published: 1 January 2009
First available in Project Euclid: 2 February 2010

zbMATH: 1243.60036

Digital Object Identifier: 10.1214/09-IMSCOLL517