The sample ACF of a simple bilinear process
type: Item
instance of: scholarly article
title: The sample ACF of a simple bilinear process
zbMATH Open document ID: 0997.60012
DOI: 10.1016/S0304-4149(99)00013-7
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: Let \(X_t\) be a bilinear process defined by \[ X_t = aX_{t-1} + bX_{t-1}Z_{t-1} + Z_t, \] where \(Z_t\) are iid random variables with standard normal distribution \(\mathcal N(0,1) \) and \(a, b \) are real constants such that \(E|a + b Z_1|^\alpha = 1\) for some \(\alpha > 0.\) Then \(X_t\) has a distribution with regularly varying tails of the same index \(\alpha\). An asymptotic distribution of the sample autocorrelations of \(X_t\) is studied in three different cases, namely for \(\alpha \in (0,2),\;\alpha \in (2,4)\) and \(\alpha > 4 .\) While in the latter case the standard central limit theory for strongly mixing conditions is used, in the first two cases a point processes methodology developed recently by \textit{R. A. Davis} and \textit{T. Hsing} [Ann. Probab. 23, No. 2, 879-917 (1995; Zbl 0837.60017)] and by \textit{R. A. Davis} and \textit{T. Mikosch} [Ann. Stat. 26, No. 5, 2049-2080 (1998; Zbl 0929.62092)] is applied to obtain limiting distributions.
reviewed by: Zuzana Prášková
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 1792527
zbMATH Keywords: bilinear process
zbMATH Keywords: sample autocorrelation function
zbMATH Keywords: heavy tails
zbMATH Keywords: point process
zbMATH Keywords: mixing conditions
author: Bojan Basrak
author: Richard A. Davis
author: T. Mikosch
MaRDI profile type: Publication
cites work: Q5525727
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full work available at URL: s0304-4149(99)00013-7
OpenAlex ID: W2038005533
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description: scientific article; zbMATH DE number 1792527