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

TL;DR

This paper establishes an almost sure limit theorem for Wick powers of Gaussian difference quotients, showing convergence to a generalized derivative under certain regularity conditions on the Gaussian process.

Contribution

It provides a new almost sure limit theorem for Wick powers of Gaussian difference quotients, extending the understanding of their asymptotic behavior.

Findings

Proves almost sure convergence of Wick powers of Gaussian difference quotients.

Identifies conditions on the Gaussian process for the limit theorem to hold.

Connects the limit to a generalized derivative of the Gaussian process.

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)= \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.