Sharp Error Terms for Return Time Statistics Under Mixing Conditions

We describe the statistics of repetition times of a string of symbols in a stochastic process. We consider a string $A$ of length $n$ and prove:1) The time elapsed until the process starting with $A$ repeats $A$, denoted by $\ta$, has a distribution which can be well approximated by a degenerated law at the origin and an exponential law.2) The number of consecutive repetitions of $A$, denoted by $S_A$, has a distribution which is approximately a geometric law.We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of $\ta$ and $S_A$. Our results hold for processes that verify the $\phi$ mixing condition.