We establish abstract limit theorems which provide sufficient conditions for a sequence of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to before some time which is of order . In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.
Nous établissons des théorèmes limites abstraits qui fournissent des conditions suffisantes pour qu’une suite d’événements rares dans un système préservant une mesure de probabilité ergodique satisfasse des asymptotiques de Poisson, et pour que les positions dans soient asymptotiquement iid (limites de Poisson spatio-temporelles). Les théorèmes limites n’utilisent que des informations sur ce qui arrive à avant un certain temps d’ordre . En particulier, nous n’utilisons aucune hypothèse sur le comportement asymptotique du système du type conditions de mélange classiques. Nous discutons également quelques questions générales sur le comportement asymptotique des processus spatiaux et spatio-temporels, et illustrons nos résultats avec des systèmes prototypiques simples.
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Keywords: invariant measure, limit distribution, rare events, Poisson process
@article{AHL_2022__5__1361_0, author = {Zweim\"uller, Roland}, title = {Hitting {Times} and {Positions} in {Rare} {Events}}, journal = {Annales Henri Lebesgue}, pages = {1361--1415}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.154}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ahl.154/} }
Zweimüller, Roland. Hitting Times and Positions in Rare Events. Annales Henri Lebesgue, Volume 5 (2022), pp. 1361-1415. doi : 10.5802/ahl.154. http://www.numdam.org/articles/10.5802/ahl.154/
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