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

Degtyar’, S. V.

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.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

ergodic Markov process; Markov functionals; asymptotic behavior

Citations:

Zbl 0927.60005