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Entry times into asymptotically receding regions for processes with semi- Markov switchings. (English. Russian original) Zbl 0568.60082

Theory Probab. Appl. 29, 558-563 (1985); translation from Teor. Veroyatn. Primen. 29, No. 3, 539-544 (1984).
A random process \((\xi (t))_{t\geq 0}\) with a measurable phase space (X,\({\mathcal B})\) is considered. Let \((\alpha_ n)^{\infty}_{n=1}\) be an increasing sequence of positive random values and for each \(D\in {\mathcal B}\) the random value \(\tau_{\alpha_ n}(D)=\inf \{s>\alpha_ n,\xi (s)\in D\}\) be defined. The sequence \((\kappa_ n,\eta_ n,\chi_ n(D))\) is assumed to be a Markov renewal process, where \(\kappa_ n=\alpha_ n-\alpha_{n-1}\), \(\eta_ n=\xi (\alpha_ n)\), \(\chi_ n(D)=1(\tau_{\alpha_{n-1}}(D)\leq \alpha_ n)\) \((\alpha_ 0=0)\). A distribution of the random value \(\tau_ 0(D_{\epsilon})\) is investigated, where \(P_ x(\tau_ 0(D_{\epsilon})\geq \alpha_ n)\to 1\) as \(\epsilon\) \(\to 0\) for each initial point \(x\in X\) and \(n\geq 1\) (asymptotically moving off condition for \((D_{\epsilon})_{\epsilon >0})\). Some conditions on \({\mathcal B}\), on the Markov chain \((\eta_ n,\chi_ n(D))\) and on \((D_{\epsilon})\) are proved to be sufficient for a normed distribution of \(\tau_ 0(D_{\epsilon})\) to converge to an exponential distribution. Also a necessary condition for the asymptotical moving off is obtained.
Reviewer: B.P.Kharlamov

MSC:

60K15 Markov renewal processes, semi-Markov processes
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