Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

Abstract

We study the asymptotic behavior of the $$\nu $$ -symmetric Riemann 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 Riemann sums converge in the Skorohod topology when $$\alpha =(2\ell +1)^{-1}$$ , where $$\ell $$ denotes the largest positive integer satisfying $$\int _{0}^{1}x^{2j}\nu (\mathrm{d}x)=(2j+1)^{-1}$$ for all $$j=0,\dots , \ell -1$$ . In the case $$\alpha >(2\ell +1)^{-1}$$ , we prove that the convergence holds in probability.