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A sufficient condition is obtained for the weak convergence of additive functionals defined on a sequence of Markov chains Xn approaching a Markov process X. The condition is expressed in terms of transient probabilities of the chains Xn. An application of the main result is given for the convergence on the Cantor set of local-time type functionals of random walks approaching an α-stable process with index α ≤ 1.

Sufficient conditions for the convergence of local-time type functionals of Markov approximations