Convergence of long-memory discrete kth order Volterra processes

We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the kth order Volterra processes of order k. These are moving average kth order polynomial forms: X(n)=∑0<i1,…,ik<∞a(i1,…,ik)ϵn−i1…ϵn−ik, where {ϵi} is i.i.d. with Eϵi=0, Eϵi2=1, where a(⋅) is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for X(n) to be well-defined in L2(Ω), 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(⋅) decays fast enough so that X(n) has short memory. We prove a non-central limit theorem if, on the other hand, a(⋅) 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.