RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Teor. Veroyatnost. i Primenen.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 5 scientific papers (total in 5 papers)

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 $a_{11}<0$,
$$a_{22}\leq0,a_{33}=0,a_{12}>0,a_{13}\geq0,a_{23}>0,a_{ij}=0,i>j.$$


Full text: PDF file (907 kB)

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

Document Type: Article
Received: 09.09.1960

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}


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

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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  mathnet  crossref  mathscinet  zmath  isi
    2. Vladimir A. Vatutin, Elena E. Dyakonova, “Extinction of decomposable branching processes”, Discrete Math. Appl., 26:3 (2016), 183–192  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  isi  elib  elib
    4. V. A. Vatutin, E. E. D'yakonova, “Decomposable branching processes with two types of particles”, Discrete Math. Appl., 28:2 (2018), 119–130  mathnet  crossref  crossref  isi  elib
    5. E. E. D'yakonova, “A subcritical decomposable branching process in a mixed environment”, Discrete Math. Appl., 28:5 (2018), 275–283  mathnet  crossref  crossref  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
    Number of views:
    This page:23
    Full text:7

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2018