On the stationary tail index of iterated random Lipschitz functions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Alsmeyer, Gerold¹ 

[1] Univ Munster, Inst Stat Math, Dept Math & Comp Sci, D-48149 Munster, Germany

关键词: Iterated function system;    Random Lipschitz function;    Mean and strongly contractive;    Stationary law;    Stochastic fixed-point equation;    Tail index;    Implicit renewal theory;    AR(1)-model with ARCH errors;    Random logistic transform;    Stochastic Ricker model;   

DOI  :  10.1016/j.spa.2015.08.004

来源: Elsevier

【 摘 要 】

Let Psi, Psi(1), Psi(2), ... be a sequence of i.i.d. random Lipschitz maps from a complete separable metric space (X, d) with unbounded metric d to itself and let X-n = Psi(n) o ... o Psi(1)(X-0) for n = 1, 2, ... be the associated Markov chain of forward iterations with initial value X-0 which is independent of the Psi(n). Provided that (X-n)(n >= 0) has a stationary law pi and picking an arbitrary reference point x(0) is an element of X, we will study the tail behavior of d(x(0), X-0) under P-pi, viz, the behavior of P-pi (d(x(0), X-0) > t) as t -> infinity, in cases when there exist (relatively simple) nondecreasing continuous random functions F, G : R->= -> R->= such that F(d(x(0), x)) <= d(x(0), Psi(x)) <= G(d(x(0), x)) for all x is an element of X and n >= 1. In a nutshell, our main result states that, if the iterations of i.i.d. copies of F and G constitute contractive iterated function systems with unique stationary laws pi(F) and pi(G) having power tails of order v(F) and v(G) at infinity, respectively, then lower and upper tail index of v = P-pi (d(x(0), X-0) is an element of.) (to be defined in Section 2) are falling in [v(G), v(F)]. If v(F) = v(G), which is the most interesting case, this leads to the exact tail index of v. We illustrate our method, which may be viewed as a supplement of Goldie's implicit renewal theory, by a number of popular examples including the AR(1)-model with ARCH errors and random logistic transforms. (C) 2015 Elsevier B.V. All rights reserved.

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