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

TL;DR

This paper investigates the heavy tail behavior of a Markov chain defined by Lipschitz maps and demonstrates convergence of normalized sums to a stable law, revealing the chain's long-term distribution characteristics.

Contribution

It establishes conditions under which the stationary measure exhibits heavy tails and proves convergence of sums to an alpha-stable law for Lipschitz iterated maps.

Findings

The stationary measure has a heavy tail with a specific tail index.

Normalized sums of the chain converge to an alpha-stable distribution.

Conditions for heavy tails and stable law convergence are explicitly characterized.

Abstract

We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{\theta_{n}}(X_{n-1}^{x})$, starting at $x\in\R^d$, where $\theta_{1}, \theta_{2},...$ are i.i.d. random variables taking their values in a metric space $(\Theta, \mathfrak{r}),$ and $\p_{\theta_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $\nu$. Under appropriate assumptions on $\p_{\theta_n}$, we will show that the measure $\nu$ has a heavy tail with the exponent $\alpha>0$ i.e. $\nu(\{x\in\R^d: |x|>t\})\asymp t^{-\alpha}$. 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 $\alpha\in(0, 2]$.