The joint probability generating function for run-lengths in regenerative binary markov chains, with applications

Gontcharov obtained the joint probability generating function for the numbers of runs of all lengths, both of successes and failures, in a Bernoulli sequence.This is here generalized to a class of regenerative binary Markov processes.For an allied class of Markov processes, the probability generating function is obtained for a "total score" defined in terms of runs of successes only, and asymptotic formulas are derived for the expectation and variance of the score.