Marchenko–Pastur law
E898463
The Marchenko–Pastur law is a probability distribution that describes the asymptotic eigenvalue spectrum of large random covariance matrices in random matrix theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Marchenko–Pastur law canonical | 3 |
| Marchenko–Pastur distribution | 1 |
| Pastur–Marchenko law | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991184 — 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: Marchenko–Pastur law Context triple: [random matrix theory, hasKeyConcept, Marchenko–Pastur law]
-
A.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
-
B.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
C.
Isserlis’ theorem in probability theory
Isserlis’ theorem in probability theory is a result that expresses higher-order moments of jointly Gaussian random variables in terms of sums of products of their pairwise covariances.
-
D.
Szegő limit theorem
The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.
-
E.
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.
- 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: Marchenko–Pastur law Target entity description: The Marchenko–Pastur law is a probability distribution that describes the asymptotic eigenvalue spectrum of large random covariance matrices in random matrix theory.
-
A.
Wigner surmise
The Wigner surmise is an approximate formula in random matrix theory that describes the statistical distribution of spacings between neighboring energy levels in complex quantum systems.
-
B.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
C.
Isserlis’ theorem in probability theory
Isserlis’ theorem in probability theory is a result that expresses higher-order moments of jointly Gaussian random variables in terms of sums of products of their pairwise covariances.
-
D.
Szegő limit theorem
The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
law in random matrix theory
ⓘ
probability distribution ⓘ |
| alsoKnownAs |
Marchenko–Pastur distribution
ⓘ
linked to:
Marchenko–Pastur law
|
| appliesTo |
Wishart matrices
ⓘ
linked to:
Wishart distribution
sample covariance matrices ⓘ |
| assumption |
entries have finite variance
ⓘ
entries have zero mean ⓘ entries of underlying random matrix are independent and identically distributed ⓘ |
| category | limiting spectral distribution ⓘ |
| convergenceType | almost sure convergence of empirical spectral distribution ⓘ |
| densityFormula | f(x) = (1 / (2π λ x)) sqrt((b - x)(x - a)) for x in [a,b] ⓘ |
| densityOutsideSupport | 0 ⓘ |
| dependsOn | limiting ratio of matrix dimension to sample size ⓘ |
| describes |
asymptotic eigenvalue distribution of large random covariance matrices
ⓘ
limiting empirical spectral distribution of eigenvalues ⓘ |
| field |
probability theory
ⓘ
random matrix theory ⓘ |
| hasAtomAtZero | yes when λ < 1 ⓘ |
| influenced |
development of modern random matrix theory
ⓘ
high-dimensional principal component analysis ⓘ |
| limitRegime | matrix dimension and sample size go to infinity with fixed ratio ⓘ |
| massAtZero | 1 - λ for λ < 1 ⓘ |
| matrixModel | X X^T where X has i.i.d. entries with zero mean and finite variance ⓘ |
| mean | 1 ⓘ |
| momentType | moments given by Narayana numbers in free probability formulation ⓘ |
| namedAfter |
Leonid Pastur
NERFINISHED
ⓘ
Vladimir Marchenko NERFINISHED ⓘ |
| noAtomAtZero | when λ ≥ 1 ⓘ |
| originalContext | asymptotic theory of random matrices with independent entries ⓘ |
| parameter | λ ⓘ |
| parameterConstraint | λ > 0 ⓘ |
| parameterType | aspect ratio of matrix dimensions ⓘ |
| relatedTo |
Wigner semicircle law
ⓘ
free Poisson distribution ⓘ free probability theory ⓘ |
| support | [a,b] subset of real numbers ⓘ |
| supportDomain | nonnegative real line ⓘ |
| supportLowerEndpoint | (1 - sqrt(λ))^2 ⓘ |
| supportType | compact support ⓘ |
| supportUpperEndpoint | (1 + sqrt(λ))^2 ⓘ |
| type | continuous distribution ⓘ |
| usedFor |
benchmarking empirical covariance eigenvalues
ⓘ
spectral analysis of high-dimensional covariance matrices ⓘ understanding eigenvalue spectra in multivariate statistics ⓘ |
| usedIn |
high-dimensional statistics
ⓘ
signal processing ⓘ statistical physics ⓘ wireless communications ⓘ |
| yearIntroduced | 1967 ⓘ |
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: Marchenko–Pastur law Description of subject: The Marchenko–Pastur law is a probability distribution that describes the asymptotic eigenvalue spectrum of large random covariance matrices in random matrix theory.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Marchenko–Pastur law
linked to: Marchenko–Pastur law