Gladyshev's Theorem for integrals with respect to a Gaussian process

Details Gladyshev's Theorem for integrals with respect to a Gaussian process Gladyshev's Theorem for integrals with respect to a Gaussian process

2011-05-08

arxiv.org/abs/1105.1503v1

Mathematics

Probability

28 pages

Scientific paper

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 normalized powers of increments of Y over a sequence of partitions of a time interval converges 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.

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