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

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.