Limit theorems for difference additive functionals (Q2890722)

The author study the limit behaviour of the following functionals NEWLINE\[NEWLINE\phi _ {n} ^ {s, t} := \sum _ {s \leq t_ {n, k} < t} F_ {n, k} (X_ {n} (t_ {n, k}))NEWLINE\]NEWLINE 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.