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arXiv 0907.2261math.PR
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重尾现象与迭代Lipschitz映射的稳定性

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

Mariusz Mirek

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

我们考虑由随机递归$X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$定义的$\R^d$上的马尔可夫链$\{X_n^x\}_{n=0}^\infty$，从$x\in\R^d$开始，其中$θ_{1}, θ_{2},...$是独立同分布的。取值于度量空间$(Θ, \mathfrak{r}),$和$\p_{θ_{n}}:\R^d\mapsto\R^d$的随机变量是Lipschitz映射。假设马尔可夫链有唯一的平稳测度PDMATHAMT。在对$\p_{θ_n}$的适当假设下，我们将证明测度$ν$具有指数为$α>0$的重尾，即$ν(\{x\in\R^d: |x|>t\})\asymp t^{-α}$。利用这一结果，我们表明，适当规范的Birkhoff和$S_n^x=\sum_{k=1}^n X_k^x$，收敛于法律的$α$ -稳定的法律$α\in(0, 2]$。

英文摘要：

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

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