Sojourn times (Q675251)

Summary: Let \(\{\zeta(u), u\geq 0\}\) be a stochastic process with state space \(A\cup B\) where \(A\) and \(B\) are disjoint sets. Denote by \(\beta(t)\) the total time spent in state \(B\) in the interval \((0,t)\). This paper deals with the problem of finding the distribution of \(\beta(t)\) and the asymptotic distribution of \(\beta(t)\) as \(t\to\infty\) for various types of stochastic processes. The main result is a combinatorial theorem which makes it possible to find in an elementary way, the distribution of \(\beta(t)\) for homogeneous stochastic processes with independent increments.