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We establish abstract limit theorems which provide sufficient conditions fora sequence $(A_{l})$ of rare events in an ergodic probability preservingdynamical system to exhibit Poisson asymptotics, and for the consecutivepositions inside the $A_{l}$ to be asymptotically iid (spatiotemporal Poissonlimits). The limit theorems only use information on what happens to $A_{l}$before some time $\tau_{l}$ which is of order $o(1/\mu(A_{l}))$. In particular,no assumptions on the asymptotic behavior of the system akin to classicalmixing conditions are used. We also discuss some general questions about theasymptotic behaviour of spatial and spatiotemporal processes, and illustrateour results in a setup of simple prototypical systems.

Hitting Times and Positions in Rare Events