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Diffusion approximation for integral functionals in the double merging and averaging scheme. (English. Ukrainian original) Zbl 0955.60041
Theory Probab. Math. Stat. 60, 87-94 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 77-84 (1999).
The authors consider the family of Markov processes with jumps \(x^{\varepsilon}(t)\), \(t\geq 0\), \(\varepsilon>0\), on a measurable space \((X, A)\) with the generating operator \(Q^{\varepsilon}\), \[ Q^{\varepsilon}\varphi(x)=q(x) \int \limits_X P^{\varepsilon}(x,dy)[\varphi (y)- \varphi(x)], \] such that the kernel \(P^{\varepsilon}(x,dy)\) may be represented in the form \(P^{\varepsilon}(x,dy)=P(x,dy)+ \varepsilon P_1(x,dy) +\varepsilon^2 P_2(x,dy),\) where \(P(x,dy)\), \(P_1(x,dy)\) and \(P_2(x,dy)\) are kernels of generating operators of certain Markov processes. The authors investigate the integral functional of the form \(\zeta^{\varepsilon}(t)= \int_0^t a(x^{\varepsilon}(s)) ds,\) where \(a(\cdot)\) is a certain function. Results on diffusion approximation of this functional are presented.

60G25 Prediction theory (aspects of stochastic processes)
60F17 Functional limit theorems; invariance principles