We prove exponential estimates for plurisubharmonic functions with respect to Monge-Amp`ere measures with H¨older continuous
potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps
on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h.
observables, and the large deviations theorem for bounded d.s.h. observables and H¨older continuous observables.