Large deviation results for waiting times in repeated experiments

Abstract

Let Z1, Z2 . . . . be a sequence of independent, identically distributed random variables with a finite set of possible values 27. Let Ai (i= 1, 2, ..., r) be finite sequences of length k over 27 and denote by zi the waiting time until At occurs as a run in the process Z1, Z2, .... We shall prove a large deviation type result on the asymptotic independence and exponentiality of the stopping times z~. The method of proof is a refinement of the reasoning applied in M6ri and Szrkely [6], where a limit theorem is proved for the waiting time for pure runs (homogenous patterns) of arbitrary events by reducing the joint distribution problem to the asymptotic exponentiality of the minimum waiting time. Our results verify the observation that the limit behaviour of extremals of waiting times for runs can be computed in the same way as extremals of independent exponentially distributed random variables. For example, if n urns are given and balls are placed at random in these urns one after another till there is at least one ball in every urn, then the number of balls needed has a double exponential limit distribution as n tends to infinity (Erd6s and Rrnyi [1]). The same limiting behaviour is found for the maximum ofn independent, exponentially distributed random variables (with expectation n). A great number of papers is devoted to the systematic study of similar problems in more general situations; here we refer only to Ivanov and Novikov [3J and the handbook of Kolchin, Sevast'yanov and Chistyakov [4].