Limit theorems for functionals of moving averages

Abstract

Let $X_n =\sum_{i=1}^\infty a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are i.i.d. with mean 0 and finite second moment and the $a_i$ are either summable or regularly varying with index $\in (-1,-1/2)$ . The sequence ${X_n}$ 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 ($\sqrt{N}$ rate) and noncentral (non-$\sqrt{N}$ rate) limit theorems for $S_N \equiv \sum_{n=1}^N [K(X_n) - EK (X_n)]$. Specifically, we show that in the short-memory case the central limit theorem holds for $S_N$ and in the long-memory case, $S_N$ 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.