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Teor. Veroyatnost. i Primenen., 1961, Volume 6, Issue 1, Pages 31–46 (Mi tvp4746)  

Transient Phenomena in Branching Stochastic Processes

V. P. Chistyakov

Moscow

Abstract: Let $\mu _k (t)=\{\mu _{k_1}(t),…,\mu _{kn}(t)\} $ be a branching process with $n$ types of particles and let
$$\mathbf P\{\mu _{kj}(t)=\omega_j,j=1,…,n\}=\delta_k^\omega+p_k^\omega t+o(t),\quad k=1, …,n,$$
when $t\to 0$. Here $\omega=\{\omega_1,…,\omega_n\},\delta_k^\omega=1$ for $\omega_k=1,\omega_j=0, j\ne k$, and $\delta_k^\omega=0$ in other cases. We define the generating functions by $f_k({x_1, …x_n})=\sum {p_k^\omega x_1^{\omega_1}…x_n^{\omega_n}},k=1,…,n,$ and denote factorial moments by
$$a_{kj}=\frac{\partial f_k}{\partial x_j}|_{x=1},\quad b_{ij}^k=\frac{\partial^2f_k}{\partial x_i\partial x_j}|_{x=1},\quad c_{ijl}^{(k)}= \frac{\partial^3f_k}{\partial x_i\partial x_j\partial x_l}|_{x=1}.$$
Let $\mathfrak{A}$ be the compact set of an undecomposable matrix $a=\|{a_{kj}}\|,k,j=1,…,n,\lambda=\max_{1\leq i\leq n}(\operatorname{Re} \lambda_i)$, where the numbers $\lambda _i$ satisfy the equality $|{a-\lambda_i E}|=0$ (Ebeing the unity matrix) and let $v=\{v_i\}_{i=1}^n,u=\{u_i\}_{i=1}^n$ satisfy the equalities
$$au= \lambda u,\quad va=\lambda v,\quad\sum\limits_{k=1}^n{v_k^2=1,}\quad\sum\limits_{k=1}^n {u_k v_k=1}.$$

Let $\mathrm K(\mathfrak{A},B,c)$ be a class of $\{f_k (x)\}$ with $a\in \mathfrak{A},0<\delta<\sum\nolimits_{i,j,k=1}{b_{ij}^{(k)}}<B<\infty,c_{i,j,l}^{(k)}<c<\infty$. The following asymptotic formula for $t\to\infty,\lambda\to0$ holds true uniformly for all $\{{f_k}\}\in\mathrm K$
$$1-\mathbf P\{\mu_{ij}(t)=0,j=1,…,n|\mu_i>0\}\sim\mu_i k(t,\lambda,0),$$
where $k(t,\lambda ,x)$ is given by (7), $\mu _i \sum\nolimits_{j=1}^n{\mu _{ij}(t)}$. The probability distributions
$$S_k^{(t)}(y_1,…y_n)=\mathbf P\{\frac{\mu_{kj}(t)}{\mathbf M\{\mu _{kj}|\mu_k<0\}}<y_j,j=1,…,n,y_j ,j=1,…,n|\mu _k > 0\}$$
converge to an exponential distribution as $t \to \infty ,\lambda \to 0$, uniformly for all $\{f_k\}\in\mathrm K$.

Full text: PDF file (1412 kB)

English version:
Theory of Probability and its Applications, 1961, 6:1, 27–41

Received: 18.11.1959

Citation: V. P. Chistyakov, “Transient Phenomena in Branching Stochastic Processes”, Teor. Veroyatnost. i Primenen., 6:1 (1961), 31–46; Theory Probab. Appl., 6:1 (1961), 27–41

Citation in format AMSBIB
\Bibitem{Chi61}
\by V.~P.~Chistyakov
\paper Transient Phenomena in Branching Stochastic Processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1961
\vol 6
\issue 1
\pages 31--46
\mathnet{http://mi.mathnet.ru/tvp4746}
\transl
\jour Theory Probab. Appl.
\yr 1961
\vol 6
\issue 1
\pages 27--41
\crossref{https://doi.org/10.1137/1106002}


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