Exact and asymptotic results for pattern waiting times

Abstract

Waiting time problems for the occurrence of a pattern have attracted considerable research interest. Several results, including Poisson or Compound Poisson approximations as well as Normal approximations have appeared in the literature. In addition, a number of asymptotic results has been developed by making use of the finite Markov chain imbedding technique and the Perron–Frobenius eigenvalue. In the present paper we present a recursive scheme for the evaluation of the tail probabilities of the waiting time for the first and r-th occurrence of a pattern. A number of asymptotic results (along with their rates of convergence) that do not require the existence of the Perron–Frobenius eigenvalue are also offered. These results cover a quite wide class of pattern waiting time problems and, in most cases, perform better than the ones using the Perron–Frobenius eigenvalue.