Probability theory and stochastic processes Theorem

Marchenko–Pastur law

Builds on Eigenvalue Independent and identically distributed random variables Convergence in distribution

Statement

Let $(Y_{ij})_{i, j \ge 1}$ be independent and identically distributed real random variables with mean $0$ and variance $1$ . Let $p = p(n)$ be positive integers with $p / n \to y \in (0, \infty)$ as $n \to \infty$ . Let $X_n$ be the $p \times n$ matrix $(Y_{ij})_{i \le p, j \le n}$ , let $S_n = \frac1n X_n X_n^{\mathsf{T}}$ , and let $\mu_n = \frac1p \sum_{k=1}^p \delta_{\lambda_k}$ , where $\lambda_1, \ldots, \lambda_p$ are the eigenvalues of $S_n$ . Let $a = (1 - \sqrt y)^2$ and $b = (1 + \sqrt y)^2$ . Then, almost surely, $\mu_n$ converges weakly to the Marchenko–Pastur distribution with ratio $y$ , the probability measure $\nu_y(dx) = \frac{1}{2\pi x y} \sqrt{(b - x)(x - a)} \, \mathbf{1}_{[a, b]}(x) \, dx + \Big(1 - \frac1y\Big)^{+} \delta_0(dx),$ where $t^{+} = \max(t, 0)$ .

Remarks

The proof can use the moment method or the Stieltjes transform. The Stieltjes transform $m(z) = \int \frac{\nu_y(dx)}{x - z}$ of the limit solves the quadratic equation $y z m^2 + (z + y - 1) m + 1 = 0$ , and inverting it gives the density. When $y > 1$ the matrix $S_n$ has rank at most $n < p$ , which explains the atom of mass $1 - 1/y$ at $0$ . The limit has mean $1$ and second moment $1 + y$ . Example: for $y = 1$ the support is $[0, 4]$ and the density is $\frac{1}{2\pi} \sqrt{(4 - x)/x}$ .

Going deeper

For a student

It is the analogue of the Wigner semicircle law for sample covariance matrices. In high-dimensional statistics it shows that the sample covariance matrix is a poor estimate of the true covariance when $p/n$ is not small. Eigenvalues of data that fall outside $[a, b]$ are read as signals that are not pure noise.

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Convergence in distribution concept
Eigenvalue concept
Independent and identically distributed random variables concept
Marchenko–Pastur law statement

Tag	`DCK63WTY5E`
URI	`m2://DCK63WTY5E`
Kind	statement, theorem
Writing	W0 stub
MSC	60B20, 15B52, 62H10
Status	This copy of the site has no build data. Run `m2 status DCK63WTY5E`.
About	Eigenvalue
Uses	Eigenvalue, Independent and identically distributed random variables, Convergence in distribution
Changed	2 November 2026

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