On convergence of semi-markov processes of multiplication with drift to a diffusion process

G. Sh. Lev

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 583-588

Résumé

A sequence of processes $Y_k(t)$, $t\ge0$, is considered, $Y_k(t)$ being of the form: $Y_k(0)=x$, $Y_k(t)$ are right continuous and $dY_k/dt=-1$ everywhere except at point $t_i^k=\sum_{l=1}^i\tau_{lk}$, where $Y_k(t_i^k)=\gamma_{ik}Y_k(t_i^k-0)$. Here $\{\tau_{ik}\}_{i=1}^\infty$, $\{\gamma_{ik}\}_{i=1}^\infty$ for any fixed $k$, are independent sequences of independent identically distributed positive random variables. It is proved that, under some restrictions on $\tau_{ik}$ and $\gamma_{ik}$, $Y_k(t)$converge to a diffusion process. The behaviour of this process as $t\to\infty$ is studied.