Factorizations of the sojourn times of semi-Markov random walks (Q2773621)

Let \(\xi_1,\xi_2,\dots\) be a sequence of independent identically distributed random variables and \(S_n=\xi_1+\cdots+\xi_n\). Given an interval \(A\) of the real axis, let \( u(A,n)=\text{Card}\{k\in[1,n]:S_k\in A\}\) be the sojourn time of the random walk \(S_k\). The authors obtain a factorization for the distribution of the functional \(u((\gamma_1,\gamma_2],n)\), \(\gamma_1<\gamma_2\). To this end, they reduce the problem to the special semi-Markov process which is controlled by a Markov chain with two states.