Kolmogorov continuity theorem

E320436

mathematical theorem theorem in probability theory

The Kolmogorov continuity theorem is a fundamental result in probability theory that provides conditions under which a stochastic process admits a modification with continuous (or Hölder-continuous) sample paths.

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All labels observed (3)

Label	Occurrences
Kolmogorov continuity theorem canonical	2
Kolmogorov–Chentsov continuity theorem	1
Kolmogorov–Chentsov theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T3037541 — 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: Kolmogorov continuity theorem
Context triple: [Andrei Kolmogorov, notableWork, Kolmogorov continuity theorem]

A. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
B. Clark–Ocone formula
The Clark–Ocone formula is a key result in stochastic calculus and Malliavin calculus that provides an explicit integral representation of square-integrable random variables with respect to Brownian motion.
C. Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
D. Malliavin calculus
Malliavin calculus is a branch of stochastic analysis that extends differential calculus to functionals of stochastic processes, particularly Brownian motion, enabling probabilistic proofs of regularity and smoothness for solutions to stochastic differential equations.
E. Kolmogorov distance
Kolmogorov distance is a statistical metric that measures the maximum difference between two cumulative distribution functions, commonly used to quantify convergence in distribution and in goodness-of-fit tests.
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: Kolmogorov continuity theorem
Target entity description: The Kolmogorov continuity theorem is a fundamental result in probability theory that provides conditions under which a stochastic process admits a modification with continuous (or Hölder-continuous) sample paths.

A. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
B. Clark–Ocone formula
The Clark–Ocone formula is a key result in stochastic calculus and Malliavin calculus that provides an explicit integral representation of square-integrable random variables with respect to Brownian motion.
C. Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
D. Malliavin calculus
Malliavin calculus is a branch of stochastic analysis that extends differential calculus to functionals of stochastic processes, particularly Brownian motion, enabling probabilistic proofs of regularity and smoothness for solutions to stochastic differential equations.
E. Kolmogorov distance
Kolmogorov distance is a statistical metric that measures the maximum difference between two cumulative distribution functions, commonly used to quantify convergence in distribution and in goodness-of-fit tests.
F. None of above. chosen

Statements (43)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in probability theory ⓘ
alsoKnownAs	Kolmogorov–Chentsov continuity theorem ⓘ linked to: Kolmogorov continuity theorem
appliesTo	random field ⓘ stochastic process ⓘ
assumes	moment bounds on increments of the process ⓘ polynomial bounds on moments of increments ⓘ
category	theorem about path regularity of stochastic processes ⓘ
concerns	Hölder continuity of sample paths ⓘ continuity of sample paths ⓘ sample path regularity ⓘ
concludes	existence of a Hölder-continuous modification of the process ⓘ existence of a continuous modification of the process ⓘ
context	real-valued stochastic processes indexed by time ⓘ stochastic processes indexed by higher-dimensional parameters ⓘ
field	probability theory ⓘ stochastic processes ⓘ
guarantees	existence of a modification with almost surely continuous paths ⓘ existence of a modification with almost surely locally Hölder-continuous paths ⓘ
hasConsequence	sample paths are almost surely uniformly continuous on compact intervals under assumptions ⓘ sample paths belong almost surely to certain Hölder spaces under assumptions ⓘ
implies	tightness of sample paths in spaces of continuous functions under suitable conditions ⓘ
involves	Hölder exponents for sample paths ⓘ moments of order greater than zero of increments ⓘ
namedAfter	Andrey Kolmogorov ⓘ linked to: Andrei Kolmogorov
provides	sufficient conditions for existence of Hölder-continuous modifications of stochastic processes ⓘ sufficient conditions for existence of continuous modifications of stochastic processes ⓘ
relatedTo	Kolmogorov extension theorem ⓘ Kolmogorov–Chentsov theorem ⓘ linked to: Kolmogorov continuity theorem
requires	bounds on expected values of powers of increments ⓘ
typeOf	continuity criterion ⓘ
usedBy	probabilists ⓘ researchers in mathematical finance ⓘ researchers in random geometry ⓘ researchers in statistical physics ⓘ stochastic analysts ⓘ
usedFor	establishing regularity of solutions to stochastic differential equations ⓘ establishing regularity of solutions to stochastic partial differential equations ⓘ proving continuity of sample paths of Brownian motion ⓘ proving continuity of sample paths of Gaussian processes ⓘ
usedIn	construction of continuous-time stochastic processes ⓘ construction of random fields with continuous sample paths ⓘ theory of Gaussian measures on function spaces ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Andrei Kolmogorov → notableWork → Kolmogorov continuity theorem ⓘ

Kolmogorov extension theorem → relatedTo → Kolmogorov continuity theorem ⓘ

Kolmogorov continuity theorem → relatedTo → Kolmogorov–Chentsov theorem ⓘ

linked to: Kolmogorov continuity theorem

Kolmogorov continuity theorem → alsoKnownAs → Kolmogorov–Chentsov continuity theorem ⓘ