Subcritical decay, and what the model leaves out
Section of Galton–Watson branching, part of Stochastic processes
Subcritical decay, and what the model leaves out
| Type | Section |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
If m<1, Markov’s inequality applies to the non-negative variable Z_n:
\mathbb{P}(Z_n>0)=\mathbb{P}(Z_n\ge 1)\le\mathbb{E} Z_n=m^n.
The bound tends to 0, which is a second proof that extinction has probability 1, and it is a concrete upper bound on survival to generation n. At criticality the same bound is 1, which says nothing. Extinction at m=1 comes from the shape of f, under the non-degeneracy assumption in the theorem, not from Markov’s inequality.
The model assumes independent, identically distributed reproduction and no limit on resources. A density-dependent population, in which a large generation produces fewer offspring per individual, is a different drift and can have a different long-run behaviour.
Used in
See also
Offspring probability generating function · Extinction criterion · Galton–Watson process · Mean and variance by generation · Extinction probability · Other offspring laws · Total family size · Generations and dependence