Events with exponentially vanishing probability have exponentially growing waiting time

A stationary ergodic process {Y_t} with distribution Q on the sequence space 𝒴^Z is examined. The probability mass Q(Y^k) decays and the recurrence time ℛ(Y^k) grows exponentially with rate ℋ(Q), the entropy rate. Rather than searching back for the first recurrence of the typical sequence Y^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^k