Marchenko–Pastur distribution

Short description: Distribution of singular values of large rectangular random matrices

Plot of the Marchenko-Pastur distribution for various values of lambda

In the mathematical theory of random matrices, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices. The theorem is named after Ukrainian mathematicians Volodymyr Marchenko and Leonid Pastur who proved this result in 1967.

If [math]\displaystyle{ X }[/math] denotes a [math]\displaystyle{ m\times n }[/math] random matrix whose entries are independent identically distributed random variables with mean 0 and variance [math]\displaystyle{ \sigma^2 \lt \infty }[/math], let

[math]\displaystyle{ Y_n = \frac{1}{n}X X^T }[/math]

and let [math]\displaystyle{ \lambda_1,\, \lambda_2, \,\dots,\, \lambda_m }[/math] be the eigenvalues of [math]\displaystyle{ Y_n }[/math] (viewed as random variables). Finally, consider the random measure

[math]\displaystyle{ \mu_m (A) = \frac{1}{m} \# \left\{ \lambda_j \in A \right\}, \quad A \subset \mathbb{R}. }[/math]

counting the number of eigenvalues in the subset [math]\displaystyle{ A }[/math] included in [math]\displaystyle{ \mathbb{R} }[/math].

Theorem. Assume that [math]\displaystyle{ m,\,n \,\to\, \infty }[/math] so that the ratio [math]\displaystyle{ m/n \,\to\, \lambda \in (0, +\infty) }[/math]. Then [math]\displaystyle{ \mu_{m} \,\to\, \mu }[/math] (in weak* topology in distribution), where

[math]\displaystyle{ \mu(A) =\begin{cases} (1-\frac{1}{\lambda}) \mathbf{1}_{0\in A} + \nu(A),& \text{if } \lambda \gt 1\\ \nu(A),& \text{if } 0\leq \lambda \leq 1, \end{cases} }[/math]

and

[math]\displaystyle{ d\nu(x) = \frac{1}{2\pi \sigma^2 } \frac{\sqrt{(\lambda_{+} - x)(x - \lambda_{-})}}{\lambda x} \,\mathbf{1}_{x\in[\lambda_{-}, \lambda_{+}]}\, dx }[/math]

with

[math]\displaystyle{ \lambda_{\pm} = \sigma^2(1 \pm \sqrt{\lambda})^2. }[/math]

The Marchenko–Pastur law also arises as the free Poisson law in free probability theory, having rate [math]\displaystyle{ 1/\lambda }[/math] and jump size [math]\displaystyle{ \sigma^2 }[/math].

Cumulative distribution function

Using the same notation, cumulative distribution function reads

[math]\displaystyle{ F_\lambda(x) =\begin{cases} \frac{\lambda-1}{\lambda} \mathbf{1}_{x\in[0, \lambda_{-})} + \left ( \frac{\lambda-1}{2\lambda} + F(x) \right ) \mathbf{1}_{x\in[\lambda_{-}, \lambda_{+})} + \mathbf{1}_{x\in[\lambda_{+}, \infty)} ,& \text{if } \lambda \gt 1\\ F(x)\mathbf{1}_{x\in[\lambda_{-}, \lambda_{+})} + \mathbf{1}_{x\in[\lambda_{+}, \infty)},& \text{if } 0\leq \lambda \leq 1, \end{cases} }[/math]

where [math]\displaystyle{ F(x) = \frac{1}{2\pi \lambda } \left ( \pi \lambda + \sigma^{-2} \sqrt{(\lambda_{+} - x)(x - \lambda_{-})} - (1+\lambda) \arctan \frac{r(x)^2-1}{2r(x)} + (1-\lambda) \arctan \frac{\lambda_- r(x)^2 - \lambda_+}{2\sigma^2(1-\lambda)r(x)} \right ) }[/math] and [math]\displaystyle{ r(x) = \sqrt{\frac{\lambda_{+}-x}{x-\lambda_{-}}} }[/math].

Moments

For each [math]\displaystyle{ k \geq 1 }[/math], its [math]\displaystyle{ k }[/math]-th moment is[math]\displaystyle{ \sum_{r=0}^{k-1}\frac{1}{r+1}\binom{k}{r}\binom{k-1}{r} \lambda^{r} = \frac 1k\sum_{r=0}^{k-1}\binom{k}{r}\binom{k}{r+1} \lambda^{r} }[/math]

Some transforms of this law

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by

[math]\displaystyle{ G_\mu(z)=\frac{z+ \sigma^2 (\lambda-1) - \sqrt{(z- \sigma^2(\lambda + 1))^2-4\lambda \sigma^4}}{2\lambda z \sigma^2}. }[/math]

Voiculescu's [math]\displaystyle{ R }[/math]-transform is given by

[math]\displaystyle{ R_\mu(z)=\frac{\sigma^2}{1-\sigma^2 \lambda z}, }[/math]

and the [math]\displaystyle{ S }[/math]-transform by

[math]\displaystyle{ S_\mu(z)=\frac{1}{\sigma^2 (1 + \lambda z)}. }[/math]

Application to correlation matrices

For the special case of correlation matrices, we know that [math]\displaystyle{ \sigma^2=1 }[/math] and [math]\displaystyle{ \lambda=m/n }[/math]. This bounds the probability mass over the interval defined by

[math]\displaystyle{ \lambda_{\pm} = \left(1 \pm \sqrt{\frac m n}\right)^2. }[/math]

Since this distribution describes the spectrum of random matrices with mean 0, the eigenvalues of correlation matrices that fall inside of the aforementioned interval could be considered spurious or noise. For instance, obtaining a correlation matrix of 10 stock returns calculated over a 252 trading days period would render [math]\displaystyle{ \lambda_+=\left(1+\sqrt{\frac{10}{252}}\right)^2\approx 1.43 }[/math]. Thus, out of 10 eigenvalues of said correlation matrix, only the values higher than 1.43 would be considered significantly different from random.

See also

• Wigner semicircle distribution
• Tracy–Widom distribution

References

• Götze, F.; Tikhomirov, A. (2004). "Rate of convergence in probability to the Marchenko–Pastur law". Bernoulli 10 (3): 503–548. doi:10.3150/bj/1089206408. 
• Marchenko, V. A.; Pastur, L. A. (1967). "Распределение собственных значений в некоторых ансамблях случайных матриц" (in ru). Mat. Sb.. N.S. 72 (114:4): 507–536. doi:10.1070/SM1967v001n04ABEH001994. Bibcode: 1967SbMat...1..457M.  Link to free-access pdf of Russian version
• Nica, A.; Speicher, R. (2006). Lectures on the Combinatorics of Free probability theory. Cambridge Univ. Press. pp. 204, 368. ISBN 0-521-85852-6. https://archive.org/details/lecturesoncombin00nica.  Link to free download Another free access site
• Zhang, W.; Abreu, G.; Inamori, M.; Sanada, Y. (2011). "Spectrum sensing algorithms via finite random matrices". IEEE Transactions on Communications 60 (1): 164–175. doi:10.1109/TCOMM.2011.112311.100721. 
• Epps, Brenden; Krivitzky, Eric M. (2019). "Singular value decomposition of noisy data: mode corruption". Experiments in Fluids 60 (8): 1–30. doi:10.1007/s00348-019-2761-y. Bibcode: 2019ExFl...60..121E.