Limit theorems for integral functionals of Hermite-driven processes

Consider a moving average process $X$ of the form $X(t)=\int_{-\infty}^t x(t-u)dZ_u$,$t\geq 0$,where$Z$isa(nonGaussian)Hermiteprocessoforder$q\geq 2$and$x:$\mathbb{R}$_+\to$\mathbb{R}$$ is sufficiently integrable. This paper investigates the fluctuations, as $T\to \mathrm{\infty }$, of integral functionals of the form $t\mapsto {\int }_0^{Tt}P(X(s))ds$, in the case where $P$ is any given polynomial function. It extends a study initiated in Tran (2018), where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.