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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kesten–Stigum theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10076759 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Kesten–Stigum theorem Context triple: [Harry Kesten, notableConcept, Kesten–Stigum theorem]
-
A.
Slepian–Wolf coding theorem
The Slepian–Wolf coding theorem is a fundamental result in information theory that characterizes the limits of lossless data compression for correlated sources encoded separately but decoded jointly.
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B.
Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
-
C.
Blum axioms
Blum axioms are a set of formal conditions introduced by Manuel Blum that rigorously define what constitutes a valid complexity measure in computational complexity theory.
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D.
Shannon–Hartley theorem
The Shannon–Hartley theorem is a fundamental result in information theory that quantifies the maximum error-free data transmission rate over a communication channel with a given bandwidth and signal-to-noise ratio.
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E.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Kesten–Stigum theorem Target entity description: 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.
-
A.
Slepian–Wolf coding theorem
The Slepian–Wolf coding theorem is a fundamental result in information theory that characterizes the limits of lossless data compression for correlated sources encoded separately but decoded jointly.
-
B.
Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
-
C.
Blum axioms
Blum axioms are a set of formal conditions introduced by Manuel Blum that rigorously define what constitutes a valid complexity measure in computational complexity theory.
-
D.
Shannon–Hartley theorem
The Shannon–Hartley theorem is a fundamental result in information theory that quantifies the maximum error-free data transmission rate over a communication channel with a given bandwidth and signal-to-noise ratio.
-
E.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
probability theory result ⓘ |
| appliesTo |
Galton–Watson branching process
ⓘ
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
ⓘ
linked to:
Galton–Watson branching process
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
ⓘ
Harry Kesten ⓘ |
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kesten–Stigum theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Kesten