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Teoriya Veroyatnostei i ee Primeneniya, 1975, Volume 20, Issue 1, Pages 162–170
(Mi tvp3002)
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This article is cited in 5 scientific papers (total in 5 papers)
Short Communications
Asymptotic properties of the probability of the degeneration after time $t$ for semi-Markov processes of multiplication
G. Sh. Lev Barnaul
Abstract:
Let $Y(t)$ be the process defined by
1) $Y(0)=x$,
2) $Y(t)=x\prod\limits_{i=1}^{\nu(t)}\gamma_i-\sum\limits_{i=1}^{\nu(t)}\tau_i\gamma_i\dots\gamma_{\nu(t)}-\gamma(t)$
where $\{\tau_i\}_1^\infty$ and $\{\gamma_i\}_1^\infty$ are independent sequences of independent identically distributed positive random variables and
\begin{gather*}
\nu(t)=\sup\biggl\{n\colon\sum_{i=1}^n\tau_i\le t\biggr\},
\\
\gamma(t)=t-\sum_{i=1}^{\nu(t)}\tau_i.
\end{gather*}
Let
\begin{gather*}
\zeta_x=\inf\{t\colon Y(t)\le0\mid Y(0)=x\},
\\
f(x,t)=\mathbf P(\zeta_x\ge t).
\end{gather*}
In the paper, asymptotic properties of $f(x,t)$ for $x>0$ as $t\to\infty$ are studied.
Received: 15.01.1974
Citation:
G. Sh. Lev, “Asymptotic properties of the probability of the degeneration after time $t$ for semi-Markov processes of multiplication”, Teor. Veroyatnost. i Primenen., 20:1 (1975), 162–170; Theory Probab. Appl., 20:1 (1975), 161–169
Linking options:
https://www.mathnet.ru/eng/tvp3002 https://www.mathnet.ru/eng/tvp/v20/i1/p162
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