Events with exponentially vanishing probability have exponentially growing waiting time
P. Algoet · 2002
A stationary ergodic process {Y/sub t/} with distribution Q on the sequence space /spl Yscr//sup Z/ is examined. The probability mass Q(Y/sup k/) decays and the recurrence time /spl Rscr/(Y/sup k/) grows exponentially with rate /spl Hscr/(Q), the entropy rate. Rather than searching back for the first recurrence of the typical sequence Y/sup k/, we wait until the first occurrence of a rare event. We prove that for if Q satisfies certain mixing conditions, then there exists a polynomially growing sequence g/sup k/.