## Analytical problems of the asymptotic behavior of Markov functionals. I.(Ukrainian, English)Zbl 1199.60317

Teor. Jmovirn. Mat. Stat. 77, 28-35 (2007); translation in Theory Probab. Math. Stat. 77, 31-38 (2008).
Let $$X(t)$$ be a homogeneous Markov ergodic process with either discrete or continuous time which takes values in a phase space $$(E,\mathcal B)$$ (see, for example, V. M. Shurenkov [Ergodic theorems and related problems, Utrecht: VSP (1998; Zbl 0927.60005)]). Denote by $$P(t, x,A), t\geq0, x\in E, A\in\mathcal B$$, the transition probability of the process $$X(t)$$ and denote by $$\pi(A),A\in\mathcal B$$ the invariant probability distribution of the process $$X(t)$$. Consider a family $$\xi_{\varepsilon}(t)$$ of Markov functionals on the process $$X(t)$$ that depends on a small parameter $$\varepsilon > 0$$ and is asymptotically degenerate. That is $$\xi_{\varepsilon}(0)=\xi(0),\varepsilon>0,$$ and $$\lim_{\varepsilon\to0}{\mathbb P}_{x,i} \{\xi_{\varepsilon}(t)\not=i \}=0$$ for all $$x\in E$$, $$i\in I$$, and $$\geq0$$. Here $${\mathbb P}_{x,i}$$ is the regular conditional probability given $$X(0) = x, \xi(0) = i$$. The author considers Markov functionals with a countable space of states $$I=\{1,2,\dots\}$$. Under some conditions he proves that ${\mathbb P}_{x,i}[\varphi(X(t)), \xi_{\varepsilon}(t)=j]-p_{ij}(u)\int_E\pi(dy)\varphi(y)\to0 , \text{as}\,\varepsilon\to0,t\to\infty, \varepsilon t\to u,$ for all $$x\in E$$ and all continuous bounded functions $$\varphi(y)$$, where $$p_{ij}(u)$$ is the $$(i,j)$$-th element of the matrix $$e^{uC}$$, $$C=\| c_{ij}\| _{i,j=1}^{\infty}$$ being a special matrix.

### MSC:

 60K15 Markov renewal processes, semi-Markov processes 60J25 Continuous-time Markov processes on general state spaces

Zbl 0927.60005
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