The sample ACF of a simple bilinear process

被引：19

作者：

Basrak, B

Davis, RA ^{[1
]}

Mikosch, T

机构：

[1] Colorado State Univ, Dept Stat, Ft Collins, CO 80523 USA

[2] Univ Groningen, Dept Math, NL-9700 AV Groningen, Netherlands

来源：

STOCHASTIC PROCESSES AND THEIR APPLICATIONS | 1999年 / 83卷 / 01期

关键词：

sample autocorrelation; sample autocovariance; heavy tails; infinite variance; stable distribution; convergence of point processes; mixing condition; stochastic recurrence equation; bilinear process;

D O I：

10.1016/S0304-4149(99)00013-7

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

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 distribution 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 behaviour of the sample autocorrelations and autocovariances of this heavy-tailed non-linear process. Of particular interest is the case when alpha < 4. If alpha is an element of (0,2) we prove that the sample autocorrelations converge 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.

引用

页码：1 / 14

页数：14

共 15 条

[1]

[Anonymous], 1992, Stochastic Stability of Markov chains

[2] ON SOME LIMIT THEOREMS SIMILAR TO ARC-SIN LAW [J].