Branching processes

Random processes

• The passing of time plays an essential part in the world which we inhabit, and consequently many applications of probability involve quantities which develop randomly as time passes. Such randomly evolving processes are called random processes or stochastic processes
• Specific processes which we shall consider in some depth are
  • branching processes
  • random walks
  • Poisson processes and related processes

A model for population growth

• We define the term nomad to be a type of hypothetical object which is able to reproduce itself according to the following rules.
  • At time , there exists a single nomad.
  • This nomad lives for a unit of time and then, at time , it dies in the act of childbirth and is replaced by a family of offspring nomads.
  • These nomads have similar biographies.
• We shall assume here that the family sizes are random variables which satisfy the following two conditions:
  • the family sizes are independent random variables each taking values in ,
  • the family sizes are identically distributed random variables with known mass function p, so that the number of children of a typical nomad has mass function for
• Such a process is called a branching process.
• we shall suppose throughout this chapter that
• We introduce some notation. The set of nomads born at time is called the generation of the branching process, and we write for the number of such nomads.

The generating-function method

• ,for the probability generating function of a typical family size
• Theorem 9.4 The probability generating functions satisfy and hence is the iterate of ,
• Theorem 9.8 The mean value of is where the mean of the family-size distribution.
• It follows by Theorem 9.8 that:
• if , the nomad population is bound to become extinct, whereas if , there is a strictly positive probability that the line of descent of nomads will continue forever
• The case when is called critical since then the mean population-size equals 1 for all time; in this case, random fluctuations ensure that the population size will take the value 0 sooner or later, and henceforth nomadkind will be extinct. # An example

this is a special case of a general result.

The probability of extinction

• Theorem 9.19 (Extinction probability theorem) The probability of ultimate extinction is the smallest non-negative root of the equation
• Theorem 9.22 (Extinction/survival theorem) Assume that . The probability of ultimate extinction satisfies if and only if the mean family-size satisfies .