On the stationary tail index of iterated random Lipschitz functions

Let Ψ,Ψ1,Ψ2,…Ψ,Ψ1,Ψ2,… be a sequence of i.i.d. random Lipschitz maps from a complete separable metric space (X,d)(X,d) with unbounded metric dd to itself and let Xn=Ψn∘⋯∘Ψ1(X0)Xn=Ψn∘⋯∘Ψ1(X0) for n=1,2,…n=1,2,… be the associated Markov chain of forward iterations with initial value X0X0 which is independent of the ΨnΨn. Provided that (Xn)n≥0(Xn)n≥0 has a stationary law ππ and picking an arbitrary reference point x0∈Xx0∈X, we will study the tail behavior of d(x0,X0)d(x0,X0) under PπPπ, viz. the behavior of Pπ(d(x0,X0)>t)Pπ(d(x0,X0)>t) as t→∞t→∞, in cases when there exist (relatively simple) nondecreasing continuous random functions F,G:R≥→R≥F,G:R≥→R≥ such that F(d(x0,x))≤d(x0,Ψ(x))≤G(d(x0,x))F(d(x0,x))≤d(x0,Ψ(x))≤G(d(x0,x)) for all x∈Xx∈X and n≥1n≥1. In a nutshell, our main result states that, if the iterations of i.i.d. copies of FF and GG constitute contractive iterated function systems with unique stationary laws πFπF and πGπG having power tails of order ϑFϑF and ϑGϑG at infinity, respectively, then lower and upper tail index of ν=Pπ(d(x0,X0)∈⋅)ν=Pπ(d(x0,X0)∈⋅) (to be defined in Section  2) are falling in [ϑG,ϑF][ϑG,ϑF]. If ϑF=ϑGϑF=ϑG, which is the most interesting case, this leads to the exact tail index of νν. 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.