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 Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 1, Pages 95–104 (Mi tvp4703)

Short Communications

Some Theorems for Branching Processes with Several Types of Particles

A. A. Savin, V. P. Chistyakov

Moscow

Abstract: Let us consider a branching process with a continuous time parameter. Suppose that there are $n$ types of particles. Let $\mu_{k1}(t),…,\mu_{kn}(t)$ be the numbers of particles of types $T_1,…,T_n$, respectively, generated by a unique particle of type $T_k$ in the time interval $[0,t]$. Let ${\mathbf a}=\|{a_{ij}}\|$ be the matrix of the first differential moments and $\lambda=\max[\operatorname{Re}\lambda_1,…,\operatorname{Re}\lambda_n]$, where $|{{\mathbf a}-\lambda _i{\mathbf E}}|=0$ (${\mathbf E}$ is a unit matrix). Theorem 1 gives an asymptotical formula for $Q_k (t)=P\{\sum\nolimits_{j=1}^n{\mu_{kj}}(t)>0\}$, when $t\to\infty$ and ${\mathbf a}$ is an arbitrary matrix. Theorem 2 gives the limit distribution for
$${\mathbf P}\{{\frac{{\mu_{k1}(t)}}{t}<y_1,\frac{{\mu_{k2}(t)}}{t}<y_2,\frac{{B\mu_{k3}(t)}}{t}<y_3}\}$$
($\beta>0$ being a certain constant) when $t\to\infty$ and $a_{11}<0$,
$$a_{22}\leq0,a_{33}=0,a_{12}>0,a_{13}\geq0,a_{23}>0,a_{ij}=0,i>j.$$

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English version:
Theory of Probability and its Applications, 1962, 7:1, 93–100

Citation: A. A. Savin, V. P. Chistyakov, “Some Theorems for Branching Processes with Several Types of Particles”, Teor. Veroyatnost. i Primenen., 7:1 (1962), 95–104; Theory Probab. Appl., 7:1 (1962), 93–100

Citation in format AMSBIB
\Bibitem{SavChi62} \by A.~A.~Savin, V.~P.~Chistyakov \paper Some Theorems for Branching Processes with Several Types of Particles \jour Teor. Veroyatnost. i Primenen. \yr 1962 \vol 7 \issue 1 \pages 95--104 \mathnet{http://mi.mathnet.ru/tvp4703} \transl \jour Theory Probab. Appl. \yr 1962 \vol 7 \issue 1 \pages 93--100 \crossref{https://doi.org/10.1137/1107008} 

• http://mi.mathnet.ru/eng/tvp4703
• http://mi.mathnet.ru/eng/tvp/v7/i1/p95

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. K. Polin, “Limit theorems for decomposable critical branching processes”, Math. USSR-Sb., 29:3 (1976), 377–392
2. Vladimir A. Vatutin, Elena E. Dyakonova, “Extinction of decomposable branching processes”, Discrete Math. Appl., 26:3 (2016), 183–192
3. V. A. Vatutin, E. E. D'yakonova, “Decomposable branching processes with a fixed extinction moment”, Proc. Steklov Inst. Math., 290:1 (2015), 103–124
4. V. A. Vatutin, E. E. D'yakonova, “Decomposable branching processes with two types of particles”, Discrete Math. Appl., 28:2 (2018), 119–130
5. E. E. D'yakonova, “A subcritical decomposable branching process in a mixed environment”, Discrete Math. Appl., 28:5 (2018), 275–283
6. V. A. Vatutin, “Uslovnaya predelnaya teorema dlya blizkikh k kriticheskim vetvyaschikhsya protsessov s finalnym tipom chastits”, Matem. vopr. kriptogr., 9:4 (2018), 53–72