Limit theorems of ergodic type for nonnegative additive functionals over Markov processes in the scheme of series (Q2739601)

Let \(\xi_{\varepsilon}^{x}(t)\), \(t\geq 0,\) for every \(\varepsilon>0\) be a homogeneous Markov process with the phase space \((X,F_X)\) and with the initial distribution concentrated at the point \(x\). Let \(g_{\varepsilon u}(x)\), \(x\in X\), \(\varepsilon>0,\) be nonnegative \(F_X\)-measurable functions and let \(\nu_{\varepsilon u}(x,t)=\int_{0}^{tT_{\varepsilon}} g_{\varepsilon u}(\xi_{\varepsilon}^{x}(v)) dv\), \(t\geq 0,\) where \(T_{\varepsilon}\to\infty\) as \(\varepsilon\to 0\). Conditions of the weak convergence of finite-dimensional distributions of the random functional \(\nu_{\varepsilon u}(x,t)\), \((u,t)\in U\times[0,\infty),\) as \(\varepsilon\to 0\) are proposed.