On the moments of “artificial” regeneration for limiting degenerated family of Markov functionals. (Ukrainian, English) Zbl 1164.60434

Dehtyar, S. V.

Teor. Jmovirn. Mat. Stat. 75, 1-7 (2006); translation in Theory Probab. Math. Stat. 75, 1-8 (2007).

Let \(X(t)\) be homogeneous aperiodic ergodic Markov process. The process \(\xi(t)\) is called Markov functional of the process \(X(t)\) if the pair \(\{X(t),\xi(t)\}\) forms a homogeneous Markov process. Let us denote the regular conditional probability under the conditions \(X(0)=x, \xi(0)=i\) by \(P_{x,i}\). The family of Markov functionals \(\xi_{\varepsilon}(t)\) of the same process \(X(t)\) such that \(\xi_{\varepsilon}(0)=\xi(0)\) is called limiting degenerated if \(\lim_{\varepsilon\to0}P_{x,i}\{\xi_{\varepsilon}(t)\neq i\}=0\) for all \(x\in E, i\in I, t\geq0\). For the process \(X(t)\) connected with limiting degenerated family \(\xi_{\varepsilon}(t)\), the author constructs the moments of “artificial” regeneration.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60K15

Markov renewal processes, semi-Markov processes

Keywords:

moments of “artificial” regeneration; limiting degenerated family of Markov functionals; homogeneous ergodic Markov process