Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

Abstract

We consider the Markov chain $\{X_n^x{\}}_{n=0}^{\infty }$ on ${\mathbb{R}}^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$, starting at $x\in {\mathbb{R}}^d$, where ${\theta }_1,{\theta }_2,...$ are i.i.d. random variables taking their values in a metric space $(\Theta ,\mathfrak{r}),$ and $\p_{θ_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $\nu$. Under appropriate assumptions on $\p_{θ_n}$, we will show that the measure $\nu$ has a heavy tail with the exponent $\alpha >0$ i.e. $\nu (\{x\in {\mathbb{R}}^d:|x|>t\})\asymp t^{-\alpha }$. Using this result we show that properly normalized Birkhoff sums $S_n^x={\sum }_{k=1}^n{X}_k^x$, converge in law to an $\alpha$--stable law for $\alpha \in (0,2]$.

23 pages, 0 figures, Accepted for publication in Probability Theory and Related Fields

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