Ergodic type conditions for frequency characteristics of non-homogeneous Markov chains (Q2759416)

Let us consider the parametric family of processes \(\nu_{\varepsilon}(x,n,t)=\sum_{k=1}^{[tT_{\varepsilon}]}g_{\varepsilon}(n+k,\eta_{\varepsilon}(x,n,k))\), where \(\varepsilon>0\); \(T_{\varepsilon}\) is a non-random positive function, \(T_{\varepsilon}\to\infty\) as \(\varepsilon\to 0\); \(g_{\varepsilon}(k,x), x\in X\), are positive \(F_{X}\)-measurable functions; \(\eta_{\varepsilon}(x,n,k)\) for each \(x\in X, n\geq 0\) is the non-homogeneous Markov chain with phase space \((X,F_{X})\), with initial distribution concentrated in the point \(x\) and transition probabilities \(P_{\varepsilon}(n+r,y,n+r+k,A)\), \(y\in X, A\in F_{X}\), depending on \(x\). For non-negative functions \(f(k,x)\), \(F_{X}\)-measurable with respect to \(x\) for any \(k\), let us denote by NEWLINE\[NEWLINER_{f}^{\varepsilon}(x,n,\lambda)=\sum_{k=0}^{\infty}e^{-\lambda kT_{\varepsilon}^{-1}}\int_{X}f(n+k,y)P_{\varepsilon}(n,x,n+k, dy),NEWLINE\]NEWLINE and denote by \(A_{\varepsilon kN}\) a sequence of measurable sets such that \(\max_{k\geq 1}\sup_{x\in A_{\varepsilon kN}}g_{\varepsilon}(k,x)=\sigma_{\varepsilon N}\to 0\) as \(\varepsilon\to 0, N\geq 0\), \(g_{\varepsilon N}^{+}(k,x)=g_{\varepsilon}(k,x)\chi_{A_{\varepsilon kN}}(x)\). The author proves that if there exist such sets \(A_{\varepsilon kN}\) and for some fixed \((z,m), z\in X, m\geq 0\), the following conditions hold true: NEWLINENEWLINENEWLINE1) \(R_{g_{\varepsilon N}^{+}}^{\varepsilon}(z,m,\lambda)\to R_{N}(\lambda)\) as \(\varepsilon\to 0\), \(\lambda>0, R_{N}(\lambda)\neq 0\); NEWLINENEWLINENEWLINE2) \( \overline{\lim\limits_{\varepsilon\to 0}}\sup\limits_{x\in A_{\varepsilon kN}, k\geq m}|R_{g_{\varepsilon N}^{+}}^{\varepsilon}(x,k,\lambda)- R_{N}(\lambda) |=0\);NEWLINENEWLINENEWLINE3) \(R_{N}(\lambda)\to R(\lambda)\) as \(N\to\infty, \lambda>0\); NEWLINENEWLINENEWLINE4) \(\lim_{\varepsilon\to 0} R_{g_{\varepsilon}}^{\varepsilon}(z,m,\lambda)= R(\lambda)\), \(\lambda>0\), where \(R(\lambda)\neq 0\), \(R_{N}(0)=R_{N}(0+0)\), \(R(0)=R(0+0)\), NEWLINENEWLINENEWLINEthen: NEWLINENEWLINENEWLINE1) \(R^{-1}(\lambda)=a+\Phi(\lambda)\), where \(a\geq 0\), \(M\exp\{-\lambda\rho(t)\}=\exp\{-t\Phi(\lambda)\}\), \(\rho(t)\) is the homogeneous non-negative process stochastically continuous from the left; NEWLINENEWLINENEWLINE2) \(\nu_{\varepsilon}(z,m,t)\Rightarrow\nu(t)=(1-\delta(t,0))\min(L_{a},\mu(t))\), \(t\in \widetilde T\), where \(\mu(t)=\inf\{s:\rho(s)>t\}, t\geq 0\), \(\widetilde T\) is the set of stochastic continuity points of \(\mu(t)\); \(L_{a}\) is exponentially distributed with parameter \(a\) random variable; the process \(\mu(t)\) and the variable \(L_{a}\) are independent.