Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Buraczewski, Dariusz¹  Damek, Ewa¹  Mirek, Mariusz¹ 

[1] Uniwersytet Wroclawski, Inst Matemat, PL-50384 Wroclaw, Poland

关键词: Markov chains;    Stationary measures;    Heavy tailed random variables;    Limit theorems;   

DOI  :  10.1016/j.spa.2011.10.010

来源: Elsevier

【 摘 要 】

Let Phi(n) be an i.i.d. sequence of Lipschitz mappings of R-d. We study the Markov chain {X-n(x)}(n=0)(infinity) on R-d defined by the recursion X-n(x) = Phi(n) (X-n-1(x)), n is an element of N, X-0(x) = x is an element of R-d. We assume that Phi(n)(x) = Phi(A(n)x, B-n(x)) for a fixed continuous function Phi : R-d x R-d -> R-d, commuting with dilations and i.i.d random pairs (A(n), B-n), where A(n) is an element of End(R-d) and B-n, is a continuous mapping of R-d. Moreover, B-n is alpha-regularly varying and A(n), has a faster decay at infinity than B-n. We prove that the stationary measure v of the Markov chain {X-n(x)} is alpha-regularly varying. Using this result we show that, if alpha < 2, the partial sums S-n(x) = Nu(n)(k=1) X-k(x), appropriately normalized, converge to an alpha-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process X-n = A(n)X(n-1) + B-n. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

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