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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 3, Pages 583–588 (Mi tvp2672)  

Short Communications

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

G. Sh. Lev

Barnaul

Abstract: 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.

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English version:
Theory of Probability and its Applications, 1973, 17:3, 551–556

Bibliographic databases:

Received: 27.01.1971

Citation: G. Sh. Lev, “On convergence of semi-markov processes of multiplication with drift to a diffusion process”, Teor. Veroyatnost. i Primenen., 17:3 (1972), 583–588; Theory Probab. Appl., 17:3 (1973), 551–556

Citation in format AMSBIB
\Bibitem{Lev72}
\by G.~Sh.~Lev
\paper On convergence of semi-markov processes of multiplication with drift to a~diffusion process
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 3
\pages 583--588
\mathnet{http://mi.mathnet.ru/tvp2672}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=305492}
\zmath{https://zbmath.org/?q=an:0299.60061}
\transl
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 3
\pages 551--556
\crossref{https://doi.org/10.1137/1117067}


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