Sharp Errors For Point-Wise Poisson Approximations In Mixing Processes

We describe the statistics of the number of occurrences of a string of symbols in a stochastic process: taking a string A of length n, we prove that the number of visits to A up to time t, denoted by N-t, has approximately a Poisson distribution. We provide a sharp error for this approximation. In contrast to previous works which present uniform error terms based on the total variation distance, our error is point-wise. As a byproduct we obtain that all the moments of Nt are finite. Moreover, we obtain explicit approximations for all of them. Our result holds for processes that verify the phi-mixing condition. The error term is explicitly expressed as a function of the rate function phi and is easily computable.