The sample ACF of a simple bilinear process

Citation
B. Basrak et al., The sample ACF of a simple bilinear process, STOCH PR AP, 83(1), 1999, pp. 1-14
Citations number
15
Categorie Soggetti
Mathematics
Journal title
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
ISSN journal
03044149 → ACNP
Volume
83
Issue
1
Year of publication
1999
Pages
1 - 14
Database
ISI
SICI code
0304-4149(19990901)83:1<1:TSAOAS>2.0.ZU;2-5
Abstract
We consider a simple bilinear process X-t = aX(t-1) + bX(t-1)Z(t-1) +Z(t), where (Z(t)) is a sequence of iid N(0, 1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207-248) that X-t has a distribu tion with regularly varying tails of index alpha > 0 provided the equation E\a + bZ(1)\(u) = 1 has the solution u = alpha. We study the limit behaviou r of the sample autocorrelations and autocovariances of this heavy-tailed n on-linear process. Of particular interest is the case when alpha < 4. If al pha is an element of (0,2) we prove that the sample autocorrelations conver ge to non-degenerate limits. If alpha is an element of (2,4) we prove joint weak convergence of the sample autocorrelations and autocovariances to non -normal limits. (C) 1999 Elsevier Science B.V. All rights reserved.