Limit theorems for additive functionals on Markov processes in the scheme of series (Q2732661)

Let \(\xi_{\varepsilon}(t)\), \(t\geq 0\), \(\varepsilon>0,\) be a homogeneous Markov process with phase space \((X, F_X)\) and transition probability \(P_\varepsilon (x,t,A)\) and let \(U\) be a parametric set. The author gives necessary and sufficient conditions for the weak convergence of the conditional finite-dimensional distributions of functionals \(\nu_{\varepsilon u}(t)\), \((u,t)\in U\times[0,+\infty)\), of the form NEWLINE\[NEWLINE \nu_{\varepsilon u}(t)=\int^{t T_\varepsilon}_0 g_{\varepsilon u}(\xi_\varepsilon (v)) dv,\quad t \geq 0, NEWLINE\]NEWLINE where \(g_{\varepsilon u}(x)\) is an \(F_X\) measurable nonnegative function, \(T_\varepsilon\) is a nonrandom nonnegative function such that \(T_\varepsilon\to+\infty\) as \(\varepsilon \to+\infty\). Conditions are given in terms of convergence of the Laplace transforms of the first order moment functions of the functionals. Similar results were proposed by the author and \textit{D. S. Sil'vestrov} [Theory Probab. Math. Stat. 41, 41-48 (1990); translation from Teor. Veroyatn. Mat. Stat. 41, 36-43 (1989; Zbl 0708.60080)].