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

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

2009-07-13

arxiv.org/abs/0907.2261v2

Mathematics

Probability

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

Scientific paper

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 $\alpha\in(0, 2]$.

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