Some refinements of limit theorems for semi-Markov processes with a general phase space. (English. Ukrainian original) Zbl 0958.60071

Jelejko, Ya. I.

Theory Probab. Math. Stat. 59, 57-65 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 57-65 (1998).

Let \(x_{t}\) be a semi-Markov process with arbitrary state space \(X\) with countably generated \(\sigma\)-algebra \(\mathcal {X}\) of sets on \(X\). Let \(x_{k}\) be the imbedded ergodic Markov chain with stationary distribution \(\Pi,\) and let \(\tau_{k}\) be moments of jumps of the process \(x_{t}.\) The author considers the following process \(\widetilde x_{t}=g(x_{\tau_{k}},t-\tau_{k})\), where \(\tau_{0}=0,\) \(x_{0}=x,\) and \(g(x,t)\) is a measurable map on \(X\times R^{+}.\) The limiting distribution for functionals \(\zeta_{t}=\int_{0}^{t}\widetilde x_{s}ds\) is found in the previous paper by the author [Ukr. Math. J. 41, No. 10, 1146-1150 (1989); translation from Ukr. Mat. Zh. 41, No. 10, 1333-1337 (1989; Zbl 0694.60083)]. The condition on the finiteness of mean values of the moments \(\tau_{k}\) is not used in the mentioned paper. The found limiting distribution exists for almost all initial states \(x\) with respect to measure \(\Pi\) of the semi-Markov process \(x_{t}\). In this paper it is shown that there exists the limiting distribution for all initial points of the semi-Markov process \(x_{t}\).

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60J25	Continuous-time Markov processes on general state spaces
60K15	Markov renewal processes, semi-Markov processes
60F15	Strong limit theorems

Keywords:

semi-Markov process with arbitrary state space; limit theorems; Laplace transform; ergodic distribution

Citations:

Zbl 0705.60076; Zbl 0694.60083