Limit theorems for functionals of moving averages

Abstract

Let Xn = Σ∞i=1aiεn-i, where the εi are i.i.d. with mean 0 and finite second moment and the ai are either summable or regularly varying with index ∈ (-1, -1/2). The sequence {Xn}has short memory in the former case and long memory in the latter. For a large class of functions K, a new approach is proposed to develop both central (√N rate) and noncentral (non-√N rate) limit theorems for SN ≡ ΣNn=1[K(Xn) - EK(Xn)]. Specifically, we show that in the short-memory case the central limit theorem holds for SN and in the long-memory case, SN can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non-central limit theorem, respectively. Further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems. Connections with the usual Hermite expansion in the Gaussian setting are also explored.