Branching process with a finite number of particle types

A model of a branching process which is a special case of a Markov process with a countable set of states. The state of the branching vector is described by the random process

$$\mu (t) = \ ( \mu _ {1} (t) \dots \mu _ {n} (t)),$$

the $k$- th component, $\mu _ {k} (t)$, of which shows that at time $t$ there are $\mu _ {k} (t)$ particles of type $T _ {k}$. The principal property by which branching processes differ from Markov processes is that the particles existing at the moment $t _ {1}$ produce daughter particles at any subsequent moment $t _ {1} + t$, $t > 0$, in a mutually independent manner. The generating functions

$$F _ {k} (t, x _ {1} \dots x _ {n} ) =$$

$$= \ {\mathsf E} (x _ {1} ^ {\mu _ {1} (t) } \dots x _ {n} ^ {\mu _ {n} (t) } \mid \mu _ {k} (0) = 1; \mu _ {i} (0) = 0, i \neq k)$$

satisfy the system of equations

$$\tag{* } F _ {k} (t + \tau , x _ {1} \dots x _ {n} ) =$$

$$= \ F _ {k} (t, F _ {1} ( \tau , x _ {1} \dots x _ {n} ) \dots F _ {n} ( \tau , x _ {1} \dots x _ {n} ))$$

with initial conditions

$$F _ {k} (0, x _ {1} \dots x _ {n} ) = x _ {k} ,\ \ k = 1 \dots n.$$

The equations (*) are satisfied by discrete-time and continuous-time processes.

In the case of discrete time, the matrix of mathematical expectations

$$A (t) = \ \| A _ {ij} (t) \| ,$$

$$\left . A _ {ij} (t) = \frac{\partial F _ {i} }{\partial x _ {j} } \right | _ {x _ {1} = \dots = x _ {n} = 1 } ,$$

is the $t$- th power of the matrix $A = A(1)$: $A(t) = A ^ {t}$. If $A$ is indecomposable and aperiodic, it has a simple positive eigen value $\lambda$ which is larger than the moduli of the other eigen values. In this case, as $t \rightarrow \infty$,

$$A _ {ij} (t) = \ u _ {i} v _ {j} \lambda ^ {t} + o ( \lambda ^ {t} ),$$

where $( u _ {1} \dots u _ {n} ), ( v _ {1} \dots v _ {n} )$ are the right and left eigen vectors of $A$ which correspond to $\lambda$. Branching processes with an indecomposable matrix $A$ are said to be subcritical if $\lambda < 1$, supercritical if $\lambda < 1$ and critical if $\lambda = 1$ and if at least one of the functions $F _ {k} (1, x _ {1} \dots x _ {n} )$ is non-linear. The concept of criticality is defined in a similar manner for continuous-time processes.

The asymptotic properties of a branching process significantly depend on its criticality. Subcritical and critical processes die out with probability one. The asymptotic formulas (as $t \rightarrow \infty$) for the probabilities,

$$Q _ {k} (t) = \ {\mathsf P} \{ \mu _ {1} (t) + \dots + \mu _ {n} (t) > 0$$

$${}\mid \mu _ {k} (0) = 1; \mu _ {i} (0) = 0, i \neq k \} ,$$

and the theorems on limit distributions of the number of particles [2], are analogous to the respective results for processes involving single-type particles (cf. Branching process). Asymptotic properties in the near-critical case ( $t \rightarrow \infty$, $\lambda \rightarrow 1$) have been studied [3]. Processes with a decomposable matrix of mathematical expectations have also been investigated [4].

References

 [1] A.N. Kolmogorov, N.A. Dmitriev, "Branching stochastic processes" Dokl. Akad. Nauk SSSR , 56 : 1 (1947) pp. 5–8 (In Russian) [2] B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) [3] V.P. Chistyakov, "On transition phenomena in branching stochastic processes with several types of particles" Theory Probab. Appl. , 17 : 4 (1972) pp. 631–639 Teor. Veroyatnost. i Primenen. , 17 : 4 (1972) pp. 669–678 [4] Y. Ogura, "Asymptotic behaviour of multitype Galton–Watson processes" J. Math. Kyoto Univ. , 15 (1975) pp. 251–302