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Limit theorems for functionals of moving averages. (English) Zbl 0903.60018

Summary: Let \(X_n= \sum^\infty_{i= 1}a_i\varepsilon_{n- i}\), 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 \(S_N\equiv \sum^N_{n= 1}[K(X_n)- EK(X_n)]\). 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 noncentral 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.

MSC:

60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes

Software:

longmemo
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