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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.

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