We proposed two asymptotic equivalent bias corrected geometric-type estimators and established the corresponding asymptotic behaviour. We applied the suggested estimators to construct asymptotic confidence intervals for this tail parameter. We proved the convergence of rare events point processes, arising from dynamical systems with fast decay of correlations. These processes count, in a certain time frame, the number of extreme events, corresponding to the entrance in some small neighbourhoods of repelling periodic points. The limit obtained was a compound Poisson process with a geometric multiplicity distribution.