Convergence of long-memory discrete $k$-th order Volterra processes

We obtain limit theorems for a class of nonlinear discrete-time processes $X(n)$ called the $k$-th order Volterra processes of order $k$. These are moving average $k$-th order polynomial forms: $$X(n)=\sum _{0<i_1,\dots ,i_k<\mathrm{\infty }}a(i_1,\dots ,i_k){ϵ}_{n-i_1}\dots {ϵ}_{n-i_k},$$ where $\{{ϵ}_i\}$ is i.i.d.\ with $\mathbb{E}{ϵ}_i=0$, $\mathbb{E}{ϵ}_i^2=1$, where $a(\cdot )$ is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for $X(n)$ to be well-defined in $L^2(\mathrm{\Omega })$, and focus on central and non-central limit theorems. We show that normalized partial sums of centered $X(n)$ obey the central limit theorem if $a(\cdot )$ decays fast enough so that $X(n)$ has short memory. We prove a non-central limit theorem if, on the other hand, $a(\cdot )$ is asymptotically some slowly decaying homogeneous function so that $X(n)$ has long memory. In the non-central case the limit is a linear combination of Hermite-type processes of different orders. This linear combination can be expressed as a centered multiple Wiener-Stratonovich integral.