Editing Branching process (section)

=== Law of large numbers for multitype branching processes ===
For multitype branching processes that the populations of different types grow exponentially, the proportions of different types converge almost surely to a constant vector under some mild conditions. This is the strong law of large numbers for multitype branching processes.

For continuous-time cases, proportions of the population expectation satisfy an [[Ordinary differential equation|ODE]] system, which has a unique attracting fixed point. This fixed point is just the vector that the proportions converge to in the law of large numbers.

The monograph by Athreya and Ney <ref>{{cite book |last1=Athreya |first1=Krishna B. |last2=Ney |first2=Peter E. |title=Branching Processes |date=1972 |publisher=Springer-Verlag |location=Berlin |isbn=978-3-642-65371-1 |pages=199–206}}</ref> summarizes a common set of conditions under which this law of large numbers is valid. Later there are some improvements through discarding different conditions.<ref>{{cite journal |last1=Janson |first1=Svante |title=Functional limit theorems for multitype branching processes and generalized Pólya urns |journal=Stochastic Processes and Their Applications |date=2003 |volume=110 |issue=2 |pages=177–245 |doi=10.1016/j.spa.2003.12.002|doi-access=free }}</ref><ref>{{cite journal |last1=Jiang |first1=Da-Quan |last2=Wang |first2=Yue |last3=Zhou |first3=Da |title=Phenotypic equilibrium as probabilistic convergence in multi-phenotype cell population dynamics |journal=PLOS ONE |date=2017 |volume=12 |issue=2 |pages=e0170916 |doi=10.1371/journal.pone.0170916 |pmid=28182672 |pmc=5300154 |bibcode=2017PLoSO..1270916J |doi-access=free }}</ref>