LIMIT-THEOREMS FOR FUNCTIONALS OF MOVING AVERAGES

Let X-n = Sigma(i=1)(infinity) alpha(i) epsilon(n-i), where the epsilo n(i) are i.i.d. with mean 0 and finite second moment and the ai are ei ther summable or regularly varying with index epsilon (- 1, - 1/2). Th e sequence {X-n} has short memory in the former case and long memory i n the latter. For a large class of functions K, a new approach is prop osed to develop both central (root n rate) and noncentral (non-root n rate) limit theorems for S-N = Sigma(n=1)(N) [K(X-n) - EK(X-n)]. Speci fically, we show that in the short-memory case the central limit theor em holds for SN and in the long-memory case, S-N can be decomposed int o two asymptotically uncorrelated parts that follow a central limit an d a noncentral limit theorem, respectively. Further we write the nonce ntral part as an expansion of uncorrelated components that follow nonc entral limit theorems. Connections with the usual Hermite expansion in the Gaussian setting are also explored.