Sharp conditions for nonexplosions and explosions in Markov jump processes (Q1897184)

A time homogeneous Markov jump process with the state space \(S \subset \mathbb{N}\) is considered; let \(T_1\), \(T_2, \ldots\) be the successive moments of jump. It is said that the process does not explode if \(T_n \to \infty\) as \(n \to \infty\), a.s. The authors give sufficient conditions for nonexplosion and explosion in terms of the parameters of the process: holding time and moments of the jump distributions.