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Short Communications

Integral equations and some limit theorems for additive functionals of Markov processes

N. I. Portenko

Donetsk

Abstract: Integral equation (3) where $V(dy)$ is a signed measure and $p(s,x,y)$ is the transition density function of a Markov process $\xi_t$ is considered. Under some conditions the solution of this equation can be considered as the characteristic function of some functional of the process
$$ \int_0^t\frac{dV}{dx}(\xi_s)\,ds $$
where $\frac{dV}{dx}(x)$ is a generalized function. Using the results obtained we prove a limit theorem for additive functionals of a sequence of sums of independent random variables with distributions tending to a stable distribution of index $\alpha$, $1<\alpha\le2$.

Received: 04.07.1966

 English version: Theory of Probability and its Applications, 1967, 12:3, 500–505

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