Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.