Approximate optimal stopping of dependent sequences

We consider optimal stopping of sequences of random variables satisfying some asymptotic independence property. Assuming that the embedded planar point processes converge to a Poisson process, we introduce some further conditions to obtain approximation of the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. This limiting problem can be solved in several cases. We apply this method to obtain approximations for the stopping of moving average sequences, of hidden Markov chains, and of max-autoregressive sequences. We also briefly discuss extensions to the case of Poisson cluster processes in the limit.