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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.