# 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 |

#### Comments

Additional references can be found in the article Branching process.

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Branching process with a finite number of particle types.

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