Hitting Times and Positions in Rare Events

We establish abstract limit theorems which provide sufficient conditions for a sequence $(A_{l})$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the $A_{l}$ to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to $A_{l}$ before some time $τ_{l}$ which is of order $o(1/μ(A_{l}))$. 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.

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