Weighted power variation of integrals with respect to a Gaussian process

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.

Published at http://dx.doi.org/10.3150/14-BEJ606 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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