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

## TL;DR

This paper investigates the asymptotic behavior of symmetric Riemann sums for functionals of self-similar Gaussian processes, establishing convergence results depending on the process's increment exponent and specific covariance conditions.

## Contribution

It provides new convergence results for symmetric Riemann sums of self-similar Gaussian processes, linking the convergence mode to the process's increment exponent and covariance structure.

## Key findings

- Weak convergence in the Skorohod topology when =(2b7+1)^{-1}
- Convergence in probability for >(2b7+1)^{-1}
- Conditions on covariance ensuring convergence modes

## 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.