  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: \[
X(n)=\sum_{0<i_1,\ldots,i_k<\infty}a(i_1,\ldots,i_k)\epsilon_{n-i_1}\ldots\epsilon_{n-i_k},
\] where $\{\epsilon_i\}$ is i.i.d.\ with $\mathbb{E} \epsilon_i=0$,
$\mathbb{E} \epsilon_i^2=1$, 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 $L^2(\Omega)$, 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.
