שמור ב:
| Main Authors: | , , |
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| פורמט: | Preprint |
| יצא לאור: |
2024
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| נושאים: | |
| גישה מקוונת: | https://arxiv.org/abs/2407.16632 |
| תגים: |
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| _version_ | 1866909265567940608 |
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| author | Chen, An Nicol, Matthew Török, Andrew |
| author_facet | Chen, An Nicol, Matthew Török, Andrew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Birkhoff sum convergence of Fréchet observables to stable laws for Gibbs-Markov systems and applications Chen, An Nicol, Matthew Török, Andrew Dynamical Systems Chaotic Dynamics 37A50, 37H99, 60F05, 60G51, 60G55 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. |
| title | Birkhoff sum convergence of Fréchet observables to stable laws for Gibbs-Markov systems and applications |
| topic | Dynamical Systems Chaotic Dynamics 37A50, 37H99, 60F05, 60G51, 60G55 |
| url | https://arxiv.org/abs/2407.16632 |