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 (1)
| Label | Occurrences |
|---|---|
| Marchenko–Pastur law canonical | 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 NERFINISHED ⓘ |
| appliesTo |
Wishart matrices
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
free Poisson distribution ⓘ free probability theory ⓘ |
| support | [a,b] subset of real numbers ⓘ |
| supportDomain | nonnegative real line GENERATED ⓘ |
| supportLowerEndpoint | (1 - sqrt(λ))^2 GENERATED ⓘ |
| supportType | compact support ⓘ |
| supportUpperEndpoint | (1 + sqrt(λ))^2 GENERATED ⓘ |
| 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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.