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.
"The Joint Probability Generating Function for Run-Lengths in Regenerative Binary Markov Chains, with Applications." Ann. Statist. 1 (5) 933 - 939, September, 1973. https://doi.org/10.1214/aos/1176342513