On the stationary tail index of iterated random Lipschitz functions

Gerold Alsmeyer

Stochastic Processes and their Applications, 2016, vol. 126, issue 1, 209-233

Abstract: Let Ψ,Ψ1,Ψ2,… 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 Xn=Ψn∘⋯∘Ψ1(X0) for n=1,2,… be the associated Markov chain of forward iterations with initial value X0 which is independent of the Ψn. Provided that (Xn)n≥0 has a stationary law π and picking an arbitrary reference point x0∈X, we will study the tail behavior of d(x0,X0) under Pπ, viz. the behavior of Pπ(d(x0,X0)>t) as t→∞, in cases when there exist (relatively simple) nondecreasing continuous random functions F,G:R≥→R≥ such that F(d(x0,x))≤d(x0,Ψ(x))≤G(d(x0,x)) for all x∈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 πF and πG having power tails of order ϑF and ϑG at infinity, respectively, then lower and upper tail index of ν=Pπ(d(x0,X0)∈⋅) (to be defined in Section 2) are falling in [ϑG,ϑF]. If ϑF=ϑG, which is the most interesting case, this leads to the exact tail index of ν. 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.

Keywords: 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 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2015.08.004

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