On the stationary tail index of iterated random Lipschitz functions

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• 作者：Gerold Alsmeyer ^{gerolda@math.uni-muenster.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60J05 ; 60H25 ; 60K05
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：126
• 期：1
• 页码：209-233
• 全文大小：305 K

文摘

Let Ψ,Ψ₁,Ψ₂,…$\Psi ,{\Psi }_1,{\Psi }_2,\dots$ be a sequence of i.i.d. random Lipschitz maps from a complete separable metric space (X,d)$(\mathbb{X},d)$ with unbounded metric d$d$ to itself and let X_n=Ψ_n∘⋯∘Ψ₁(X₀)$X_n={\Psi }_n\circ \cdots \circ {\Psi }_1\left(X_0\right)$ for n=1,2,…$n=1,2,\dots$ be the associated Markov chain of forward iterations with initial value X₀$X_0$ which is independent of the Ψ_n${\Psi }_n$. Provided that (X_n)_n≥0${\left(X_n\right)}_{n\ge 0}$ has a stationary law π$\pi$ and picking an arbitrary reference point x₀∈X$x_0\in \mathbb{X}$, we will study the tail behavior of d(x₀,X₀)$d(x_0,X_0)$ under P_π${\mathbb{P}}_{\pi }$, viz. the behavior of P_π(d(x₀,X₀)>t)${\mathbb{P}}_{\pi }(d(x_0,X_0)>t)$ as t→∞$t\to \infty$, in cases when there exist (relatively simple) nondecreasing continuous random functions F,G:R_≥→R_≥$F,G:{\mathbb{R}}_{\ge }\to {\mathbb{R}}_{\ge }$ such that for all x∈X$x\in \mathbb{X}$ and n≥1$n\ge 1$. In a nutshell, our main result states that, if the iterations of i.i.d. copies of F$F$ and G$G$ constitute contractive iterated function systems with unique stationary laws π_F${\pi }_F$ and π_G${\pi }_G$ having power tails of order ϑ_F${\vartheta }_F$ and ϑ_G${\vartheta }_G$ at infinity, respectively, then lower and upper tail index of 2252c0e9bb3d722e77573c6078653eb9" title="Click to view the MathML source">ν=P_π(d(x₀,X₀)∈⋅)$\nu ={\mathbb{P}}_{\pi }(d(x_0,X_0)\in \cdot )$ (to be defined in Section 2) are falling in [ϑ_G,ϑ_F]$[{\vartheta }_G,{\vartheta }_F]$. If ϑ_F=ϑ_G${\vartheta }_F={\vartheta }_G$, which is the most interesting case, this leads to the exact tail index of 225962dffc5460c6446f9c6589c" title="Click to view the MathML source">ν$\nu$. 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.