Lévy measure
E1020436
A Lévy measure is a mathematical tool used in probability theory to characterize the jump behavior of Lévy processes and more general infinitely divisible distributions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Lévy triplet | 3 |
| Lévy measure canonical | 2 |
| Lévy–Khintchine formula | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T13070794 — 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: Lévy measure Context triple: [Paul Lévy, knownFor, Lévy measure]
-
A.
Lévy–Prokhorov metric
The Lévy–Prokhorov metric is a probability metric on the space of probability measures that metrizes weak convergence and is widely used in probability theory and measure theory.
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B.
Liouville measure
Liouville measure is a canonical volume measure on phase space in Hamiltonian mechanics and symplectic geometry that remains invariant under the system’s time evolution.
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C.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
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D.
Radon measure
A Radon measure is a type of measure on a topological space that is locally finite and inner regular, playing a central role in modern measure theory and integration.
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E.
Stieltjes measure
A Stieltjes measure is a measure on the real line constructed from a nondecreasing, right-continuous function, providing the measure-theoretic foundation for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals.
- 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: Lévy measure Target entity description: A Lévy measure is a mathematical tool used in probability theory to characterize the jump behavior of Lévy processes and more general infinitely divisible distributions.
-
A.
Lévy–Prokhorov metric
The Lévy–Prokhorov metric is a probability metric on the space of probability measures that metrizes weak convergence and is widely used in probability theory and measure theory.
-
B.
Liouville measure
Liouville measure is a canonical volume measure on phase space in Hamiltonian mechanics and symplectic geometry that remains invariant under the system’s time evolution.
-
C.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
D.
Radon measure
A Radon measure is a type of measure on a topological space that is locally finite and inner regular, playing a central role in modern measure theory and integration.
-
E.
Stieltjes measure
A Stieltjes measure is a measure on the real line constructed from a nondecreasing, right-continuous function, providing the measure-theoretic foundation for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
measure in probability theory ⓘ |
| appearsIn |
Lévy–Khintchine formula
ⓘ
characteristic exponent of Lévy processes ⓘ generator of Lévy processes ⓘ |
| associatedWith |
Lévy triplet
ⓘ
linked to:
Lévy measure
|
| characterizes |
jump behavior of Lévy processes
ⓘ
jump intensity ⓘ jump size distribution ⓘ |
| componentOf |
Lévy triplet
ⓘ
linked to:
Lévy–Khintchine formula
|
| constraint |
integrability condition near 0 via (1 ∧ |x|^2)
ⓘ
ν({x: |x|>1}) < ∞ for many Lévy processes ⓘ |
| determines |
distribution of jumps of a Lévy process
ⓘ
infinitely divisible law ⓘ |
| domain | ℝ^d \ {0} ⓘ |
| field |
measure theory
ⓘ
probability theory ⓘ stochastic processes ⓘ |
| generalizes | intensity measure of a Poisson process ⓘ |
| namedAfter | Paul Lévy ⓘ |
| property |
measure of {0} equals 0
ⓘ
σ-finite measure ⓘ ∫_{ℝ^d \ {0}} (1 ∧ |x|^2) ν(dx) < ∞ ⓘ |
| relatedConcept |
Lévy process
ⓘ
linked to:
Lévy processes
Lévy–Itô decomposition ⓘ characteristic function ⓘ infinitely divisible measure ⓘ |
| relatedTo |
Poisson random measure
ⓘ
compound Poisson process ⓘ jump kernel in integro-differential operators ⓘ jump measure of a process ⓘ |
| role |
encodes frequency of jumps of different sizes
ⓘ
separates small and large jumps in Lévy–Itô decomposition ⓘ |
| symbol |
Π
ⓘ
ν ⓘ |
| usedIn |
CGMY processes
ⓘ
linked to:
Lévy processes
Lévy process theory ⓘ
linked to:
Lévy processes
financial mathematics ⓘ infinitely divisible distributions ⓘ jump process modeling ⓘ risk theory ⓘ stable distributions ⓘ stochastic calculus with jumps ⓘ tempered stable processes ⓘ variance gamma processes ⓘ |
| usedToDefine |
jump-diffusion process
ⓘ
pure-jump Lévy process ⓘ subordinator ⓘ |
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: Lévy measure Description of subject: A Lévy measure is a mathematical tool used in probability theory to characterize the jump behavior of Lévy processes and more general infinitely divisible distributions.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lévy measure
linked to: Lévy measure
linked to: Lévy measure
linked to: Lévy measure
linked to: Lévy measure