Sharp Error Terms for Poisson Statistics Under Mixing Conditions: A New Approach

We describe the statistics of the number of occurrences of a string of symbols in a stochastic process: Chosen 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. Contrarily to previous works who presente uniform error terms based on the total variation distance, our error is point-wise. As a byproduct we obtain approximations for all the moments of $N_t$. Our result holds for processes that verify the \phi$-mixing condition. The error term is explcitely expressed as function of $\phi$ and then easily computable. We breafly extend our result to the weaker $\alpha$-mixing case.