Geodesic
Weak convergence of additive functionals of a sequence of
Teoriâ slučajnyh processov, Tome 15 (2009) no. 1, pp. 15-32.

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We consider additive functionals $\phi_n$, $n\geq1$ defined on a sequence of Markov chains that weakly converges to a Markov process. We give sufficient condition for $\phi_n$, $n\geq1$ to converge in distribution, formulated in the terms of their characteristics (i.e. expectations). This condition generalizes Dynkin's theorem on convergence of $W$-functionals of a time homogeneous Markov process.
Keywords: Additive functional, characteristic of additive functional, $W$-functional, local time, Markov approximation. 
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Yuri. I. Kartashov; Alexey. M. Kulik. Weak convergence of additive functionals of a sequence of. Teoriâ slučajnyh processov, Tome 15 (2009) no. 1, pp. 15-32. http://geodesic.mathdoc.fr/item/THSP_2009_15_1_a2/

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