Distributions of statistics describing concentration of runs in non homogeneous Markov-dependent trials

In the first n, n ? 3, trials of a non homogeneous zero-one Markov chain of first order, we consider runs of ones of length exceeding a threshold. The article deals with statistics denoting, the length and the position of the shortest segment of the chain in which all such runs of ones are concentrated. The study provides recursive schemes for conditional distributions of these statistics. Numerical examples illustrate the theoretical results.