Kolmogorov continuity theorem
E320436
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.
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).
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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.
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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.
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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.
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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 continuity theorem
ⓘ
surface form:
Kolmogorov–Chentsov 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 |
Andrei Kolmogorov
ⓘ
surface form:
Andrey 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 continuity theorem self-linksurface differs ⓘ
surface form:
Kolmogorov–Chentsov 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
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Instruction
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Input
Subject: Kolmogorov continuity theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
Kolmogorov continuity theorem
→
relatedTo
→
Kolmogorov continuity theorem
self-linksurface differs
ⓘ
this entity surface form:
Kolmogorov–Chentsov theorem
this entity surface form:
Kolmogorov–Chentsov continuity theorem