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Limit theorems for difference additive functionals. (English. Russian original) Zbl 1247.60118

Theory Probab. Math. Stat. 83, 83-94 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 69-79 (2010).
The author study the limit behaviour of the following functionals \[ \phi _ {n} ^ {s, t} := \sum _ {s \leq t_ {n, k} < t} F_ {n, k} (X_ {n} (t_ {n, k})) \] where \(\lambda _ {n} := \{ t_ {n, k} \mid n , k \geq 1\}\) is a sequence of partitions of \(\mathbb R ^ {+}\); \(X_ {n}\) is a sequence of processes, weakly convergent to a Markov process, assuming values in a locally compact metric space \({\mathcal X}\) ; and \(F_ {n, k}\) are non-negative Borel functions defined on \({\mathcal X}\). Sufficient conditions for the convergence of the functionals, in terms of conditions for the convergence of their characteristics, under generalized conditions for the convergence of processes, are obtained. Sufficient conditions for the uniform convergence of these functionals are also proved.

MSC:

60J55 Local time and additive functionals
60J45 Probabilistic potential theory
60F17 Functional limit theorems; invariance principles
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