On the Existence of a Small Parameter for a Semi-Markov Process

Abstract

The problem of existence of a small parameter for a decomposable semi-Markov process with a finite number of states E is investigated in the case where the phase space E of the process can be represented in the form of the union $E = \bigcup\nolimits_{k = 1}^r {E_k }$ of disjoint sets E ₁,...,E_r of ergodic states. The asymptotic behavior of transition probabilities of the semi-Markov process with phase space $E = \bigcup\nolimits_{k = 0}^r {E_k }$, where E ₀ is the set of unessential states and E _k, k=1,...,r, are classes of ergodic states, is studied.