arXiv:1105.1503v1 [math.PR] 8 May 2011 Gladyshev’s Theorem for integrals with respect to a Gaussian process ∗ Rimas Norvaiša Vilnius University, Institute of Mathematics and Informatics, Akademijos 4, Vilnius, Lithuania. E-mail: rimas.norvaisa@mii.vu.lt 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 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. Keywords : covariance, double Riemann-Stieltjes integral, Gaussian process, locally stationary increments, Orey index, power variation, p-variation, quadratic mean integral Running title: Gladyshev’s Theorem for integrals 1 Introduction We consider a stochastic process Y = {Y (t): t ∈ [0,T ]}, 0 <T< ∞, given by an integral Y (t)= q.m.  t 0 fdX, 0 ≤ t ≤ T, (1) defined as a limit of Riemann-Stieltjes sums converging in quadratic mean, where f : [0,T ] → R is a real-valued deterministic function and X = {X (t): t ∈ [0,T ]} is a second order stochastic process. The process is well defined under the hypotheses on f and X stated by Theorems 9 and 10 below. More specifically, we are interested in the case when X is a Gaussian process, which may not have stationary increments and is a member of a class of processes defined as follows. * This research was funded by a grant (No. MIP-66/2010) from the Research Council of Lithuania 1