Events with exponentially vanishing probability have exponentiallygrowing waiting time

摘要

A stationary ergodic process {Y_t} with distribution Qon the sequence space 𝒴Z is examined. The probabilitymass Q(Yk) decays and the recurrence timeℛ(Yk) grows exponentially with rate ℋ(Q), theentropy rate. Rather than searching back for the first recurrence of thetypical sequence Yk, we wait until the first occurrence of arare event. We prove that for if Q satisfies certain mixing conditions,then there exists a polynomially growing sequence gk