Abstract
We consider the Markov chain \({\{X_n^x\}_{n=0}^\infty}\) on \({\mathbb{R}^d}\) defined by the stochastic recursion \({X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}\), starting at \({x\in\mathbb{R}^d}\), where θ 1, θ 2, . . . are i.i.d. random variables taking their values in a metric space \({(\Theta, \mathfrak{r})}\), and \({\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}\) are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ν. Under appropriate assumptions on \({\psi_{\theta_n}}\), we will show that the measure ν has a heavy tail with the exponent α > 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 α-stable law for \({\alpha\in(0, 2]}\).
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In memory of Andrzej Hulanicki.
This research project has been partially supported by Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389) and by KBN grant N201 012 31/1020.
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Mirek, M. Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Theory Relat. Fields 151, 705–734 (2011). https://doi.org/10.1007/s00440-010-0312-9
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DOI: https://doi.org/10.1007/s00440-010-0312-9