Asymptotic Properties of Serial Covariances for Nonlinear Stationary Processes

✍ Scribed by K.C Chanda

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 266 KB
• Volume: 47
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1993.1077

✦ Synopsis

Let (\left{X_{t} ; t \in \mathbb{Z}\right}) be a strictly stationary process with mean zero and autovariance function (a.c.v.f.) (\gamma_{x}, v \in \mathbb{Z}). Let (\hat{\gamma}{v}=n^{-1} \sum{t=1}^{n-\mid v_{i}} X_{1} X_{r+|x|}) be the serial covariance of order (v) computed from a sample (X_{1}, \ldots, X_{n}) drawn from (\left{X_{i}\right}). We assume that (\left{X_{i}\right}) is nonlinear but satisfies some mild regularity conditions. We prove that for a fixed integer (l), the distribution of (n^{1 / 2}\left(\hat{\gamma}{v}-\gamma{v}\right), \ldots, n^{1 / 2}\left(\hat{\gamma}{v+1}-\gamma{v+1}\right)) is, asymptotically, normal with mean zero and a finite covariance matrix. The result holds both for finite (v) and when (v \rightarrow \infty) but (v / n \rightarrow 0) as (n \rightarrow \infty). 1993 Academic Press, Inc.