# Branching Process

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## Branching Processes

In a simple branching process we assume that each individual produces a number of offspring, who in turn produce their own offspring. The process always starts with one individual (the #~{ancestor}) which is denoted generation 0. The offspring of the ancestor are called the 1^{st} generation, and in general the offspring of the ~n^{th} generation are called the ~n+1^{th} generation. (Note that in the simple model we take no account of the time between generations.)

## Galton-Watson Model

We make the following assumptions in the model we are dealing with:

1. The number of offspring for each individual is given by identically distributed random variables.
2. The number of offspring of any individual is independent of the number of offspring of any other.
3. The process starts with one individual (in generation 0).

This is known as the #~{Gaton-Watson} model for a branching process.

Let ~X denote the number of offspring of an individual, and put _ ~p( ~x ) _ = _ ~P ( ~X = ~x ) , _ called the #~{offspring probability function} of the individual. We have _ &sum._{~x}^{&infty.}_{= 0} _ ~p( ~x ) _ = _ 1.

The number of individuals in the ~n^{th} generation is denoted ~Z_~n. We have _ ~Z_0 = 1 , _ ~Z_1 = ~X , _ and

~Z_~n _ = _ sum{~X_~i^{( ~n - 1 )},~i = 1,~Z_{~n - 1}} _ _ _ _ ~n > 1

Where _ ~X_~i^{( ~n - 1 )} is the number of offspring of the ~i^{th} individual in the ~n - 1^{th} generation.