Symmetric stochastic integrals with respect to a class of self-similar Gaussian processes

Abstract

We study the asymptotic behavior of the $\nu$-symmetric Riemman sums for functionals of a self-similar centered Gaussian process $X$ with increment exponent $0<\alpha <1$. We prove that, under mild assumptions on the covariance of $X$, the law of the weak $\nu$-symmetric Riemman sums converge in the Skorohod topology when $\alpha =(2\ell +1{)}^{-1}$, where $\ell$ denotes the smallest positive integer satisfying ${\int }_0^1{x}^{2j}\nu (dx)=(2j+1{)}^{-1}$ for all $j=0,\dots ,\ell -1$. In the case $\alpha >(2\ell +1{)}^{-1}$, we prove that the convergence holds in probability.