Kesten–Stigum theorem
E839311
The Kesten–Stigum theorem is a fundamental result in branching process theory that characterizes when a suitably normalized supercritical branching process converges to a non-degenerate limit.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
probability theory result ⓘ |
| appliesTo |
Galton–Watson branching process
NERFINISHED
ⓘ
supercritical branching process ⓘ |
| assumes |
finite expectation of offspring number times log plus of offspring number
ⓘ
integrability condition on offspring distribution ⓘ mean offspring number greater than 1 ⓘ supercriticality ⓘ |
| characterizes | convergence of normalized population size ⓘ |
| concerns | non-degenerate limit of branching processes ⓘ |
| concludes |
almost sure convergence of Zₙ / mⁿ
ⓘ
limit random variable is non-degenerate under moment condition ⓘ limit random variable is positive with positive probability ⓘ martingale convergence of normalized population size ⓘ |
| condition | E(X log⁺ X) < ∞ ⓘ |
| dateOfFormulation | 1960s ⓘ |
| distinguishes | degenerate and non-degenerate martingale limits ⓘ |
| field |
branching process theory
ⓘ
probability theory ⓘ |
| generalizes | earlier results on branching process limits ⓘ |
| hasConsequence |
classification of survival behavior of branching processes
ⓘ
criteria for survival with positive probability ⓘ |
| hasVersion |
continuous-time branching process analogues
ⓘ
discrete-time branching process version ⓘ multi-type branching process version ⓘ |
| implies | non-trivial limit for normalized population size under moment condition ⓘ |
| involvesParameter | offspring mean m ⓘ |
| involvesRandomVariable |
generation size Zₙ
ⓘ
limit random variable W ⓘ |
| isCriterionFor | non-degeneracy of branching process limit ⓘ |
| isRelatedTo |
Galton–Watson theorem
NERFINISHED
ⓘ
branching martingale limit theorems ⓘ |
| isUsedIn |
epidemic modeling
ⓘ
information theory on trees ⓘ population biology models ⓘ random trees analysis ⓘ theory of branching processes in random environments ⓘ |
| namedAfter |
Boris Stigum
NERFINISHED
ⓘ
Harry Kesten NERFINISHED ⓘ |
| typicalStatement |
If E(X log⁺ X) < ∞ then Zₙ / mⁿ converges almost surely to a non-degenerate limit
ⓘ
If E(X log⁺ X) = ∞ then Zₙ / mⁿ converges almost surely to 0 ⓘ |
| usesConcept |
almost sure convergence
ⓘ
conditional expectation ⓘ martingale ⓘ offspring distribution ⓘ supercritical Galton–Watson process ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.