Limit theorems for functionals of moving averages

Abstract

Let $X_n =\\sum_{i=1}^\\infty a_i \\varepsilon_{n-i}$, where the\n$\\varepsilon_i$ are i.i.d. with mean 0 and finite second moment and the $a_i$\nare either summable or regularly varying with index $\\in (-1,-1/2)$ . The\nsequence ${X_n}$ has short memory in the former case and long memory in the\nlatter. For a large class of functions $K$, a new approach is proposed to\ndevelop both central ($\\sqrt{N}$ rate) and noncentral (non-$\\sqrt{N}$ rate)\nlimit theorems for $S_N \\equiv \\sum_{n=1}^N [K(X_n) - EK (X_n)]$. Specifically,\nwe show that in the short-memory case the central limit theorem holds for $S_N$\nand in the long-memory case, $S_N$ can be decomposed into two asymptotically\nuncorrelated parts that follow a central limit and a non-central limit theorem,\nrespectively. Further we write the noncentral part as an expansion of\nuncorrelated components that follow noncentral limit theorems. Connections with\nthe usual Hermite expansion in the Gaussian setting are also explored.