Weighted power variation of integrals with respect to a Gaussian process

TL;DR

This paper proves almost sure convergence of weighted power sums of increments of integrals with respect to Gaussian processes, including fractional Brownian motion, based on the covariance function's p-variation.

Contribution

It establishes convergence results for sums of powers of increments of integrals driven by general Gaussian processes, extending previous work to non-stationary cases.

Findings

Convergence holds for fractional, subfractional, and bifractional Brownian motions.

Conditions are expressed via the p-variation of the covariance function.

Results apply to a broad class of Gaussian processes without stationary increments.

Abstract

We consider a stochastic process $Y$ defined by an integral in quadratic mean of a deterministic function $f$ with respect to a Gaussian process $X$, which need not have stationary increments. For a class of Gaussian processes $X$, it is proved that sums of properly weighted powers of increments of $Y$ over a sequence of partitions of a time interval converge almost surely. The conditions of this result are expressed in terms of the $p$-variation of the covariance function of $X$. In particular, the result holds when $X$ is a fractional Brownian motion, a subfractional Brownian motion and a bifractional Brownian motion.