Gibbs-Markov系统的Fréchet观测值Birkhoff和向稳定律的收敛及应用
Birkhoff sum convergence of Fréchet observables to stable laws for Gibbs-Markov systems and applications
AI总结：
本文利用Poisson点过程方法证明了Gibbs-Markov映射上非平方可积Fréchet观测值向稳定律的分布收敛，并验证了关键混合条件，最后探讨了其在间歇型映射中的应用。
摘要：
我们使用Poisson点过程方法，证明了对非平方可积观测值$ϕ: [0,1]\to R$（主要为$ϕ(x) = d(x,x_0)^{-\frac{1}α}$,$0<α\le 2$）在Gibbs-Markov映射上向稳定律的分布收敛。一个关键结果是验证了标准混合条件，确保在$1<α\le 2$范围内，观测值的大值主导时间序列。动力系统观测值的稳定极限律已在两种设定下建立：“好观测值”（通常为Hölder）在慢混合非一致双曲系统上，以及“坏观测值”（具肥尾的无界函数）在快混合动力系统上。作为应用，我们研究了一类间歇型映射中这两种效应的相互作用。
英文摘要：
We use a Poisson point process approach to prove distributional convergence to a stable law for non square-integrable observables $ϕ: [0,1]\to R$, mostly of the form $ϕ(x) = d(x,x_0)^{-\frac{1}α}$,$0<α\le 2$, on Gibbs-Markov maps. A key result is to verify a standard mixing condition, which ensures that large values of the observable dominate the time-series, in the range $1<α\le 2$. Stable limit laws for observables on dynamical systems have been established in two settings: ``good observables'' (typically Hölder) on slowly mixing non-uniformly hyperbolic systems and ``bad'' observables (unbounded with fat tails) on fast mixing dynamical systems. As an application we investigate the interplay between these two effects in a class of intermittent-type maps.
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