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

N. I. Portenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 551-559

Résumé

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 $\alpha$, $1\alpha\le2$.