Marčenko–Pastur Law with Variance Parameter

• Marčenko–Pastur Law with Variance Parameter is a framework that defines the asymptotic eigenvalue distribution of large sample covariance matrices using the ratio y = p/n.
• The law provides explicit formulas for global spectral density and fluctuations, employing techniques like Stieltjes transforms, martingale decompositions, and polynomial approximations.
• It has significant applications in PCA, hypothesis testing, and signal processing, and extends to settings with arbitrary variance profiles and heavy-tailed distributions.

The Marčenko–Pastur law with variance parameter describes the limiting eigenvalue distribution of large sample covariance matrices in high-dimensional regimes, incorporating the effects of the underlying variance structure into both the global spectral density and the fluctuation behavior of linear spectral statistics. This law plays a central role in random matrix theory, multivariate statistics, signal processing, and the theory of high-dimensional inference. Its key feature is the explicit dependence on a “variance” parameter—typically denoted by the ratio $y = p/n$ between the matrix dimension $p$ and the sample size %%%%2%%%%—but may be generalized to include arbitrary variance profiles, higher-order moment information, and functional extensions involving more complex functionals.

1. Definition and Canonical Form of the Marčenko–Pastur Law with Variance Parameter

The classical Marčenko–Pastur (MP) law arises as the asymptotic spectral density of the eigenvalues of sample covariance matrices of the form $W = \frac{1}{p} X X^*$, where $X$ is an $n \times p$ random matrix with independent entries of mean zero and variance one. In the high-dimensional limit with $p,n \to \infty$ and $y = \frac{p}{n}$ fixed in $(0,1]$, the empirical spectral distribution $F_n(x)$ of $W$ converges to a deterministic law with density

$$f_y(x) = \frac{1}{2\pi x y} \sqrt{(b - x)(x - a)} \cdot \mathbf{1}_{[a,b]}(x)$$

supported on $[a, b]$ with

$a = (1-\sqrt{y})^2, \qquad b = (1+\sqrt{y})^2.$

The parameter $y$—which serves as a variance or aspect ratio parameter—controls both the support and the shape of the density (Bai et al., 2010, Götze et al., 2011).

2. Variance Parameter: Roles, Interpretation, and Generalizations

The variance parameter $y$ appears in several critical aspects:

• Support of the Spectrum: The endpoints $a$ and $b$ scale as functions of $y$, determining the interval on which the eigenvalues concentrate.
• Normalization: The sample covariance matrix is typically normalized by $1/p$ so that the variance of the entries sets the proper scale for the limiting law. If the entries had variance $\sigma^2$ instead of $1$, then the support would be $[\sigma^2 a, \sigma^2 b]$.
• Shaping Density and Fluctuations: The shape of the MP density and the fluctuation formulas for linear spectral statistics explicitly depend on $y$. Additionally, higher-order moment corrections and functional central limit theorems for statistics such as $\sum_j f(\lambda_j)$ involve $y$ inside their leading terms (Bai et al., 2010).

In generalized settings, the variance parameter may be replaced by a variance profile $S = (s_{ik})$, possibly non-uniform and possibly even non-primitive, leading to self-consistent (Dyson or quadratic vector) equations for the limiting density (Alt et al., 2016, Ajanki et al., 2013).

3. Functional Central Limit Theorem and Explicit Mean/Covariance Formulas

The law governs not only the global density, but also the asymptotic fluctuations of linear spectral statistics (LSS), i.e., centered sums $\sum_j f(\lambda_j)$. For test functions $f$ with sufficient smoothness (specifically, $C^4$ regularity), the LSS process

$$G_n(f) = \sum_{j=1}^p f(\lambda_j) - p \int_a^b f(x) f_y'(x) dx$$

converges to a Gaussian process $G(f)$, with mean and covariance functions (Bai et al., 2010): $$\mathbb{E} G(f) = \frac{\kappa_1}{2\pi} \int_a^b f'(x) \arg(1 - y k(x)^2) dx - \frac{\kappa_2}{\pi} \int_a^b f(x) \Im \frac{y k(x)^3}{1 - y k(x)^2} dx,$$

$$\operatorname{Cov}(G(f), G(g)) = \frac{\kappa_1 + 1}{2\pi^2} \iint_{[a,b]^2} f'(x_1) g'(x_2) \ln \left| \frac{\bar{s}(x_1) - s(x_2)}{s(x_1) - s(x_2)} \right| dx_1 dx_2 - \frac{\kappa_2 y}{2\pi^2} \iint_{[a,b]^2} f'(x_1) g'(x_2) \Re [k(x_1)k(x_2) - \bar{k}(x_1)k(x_2)] dx_1 dx_2.$$

Here, the functions $k(x)$, $s(x)$, and $\bar{s}(x)$ are defined in terms of the Stieltjes (companion) transform of the MP law, and the parameters

$$\kappa_1 = |\mathbb{E} x_{11}^2|^2, \quad \kappa_2 = \mathbb{E}|x_{11}|^4 - \kappa_1 - 2$$

encode higher-moment information.

These formulas reveal directly how the variance parameter $y$ and higher moments shape the LSS fluctuations—contributions such as $y k^2(x)$ and denominators $(1 - y k^2(x))$ capture the variance's nonlinear effects, highlighting the sensitivity of the spectral fluctuations to dimensionality and entry distribution (Bai et al., 2010).

4. Methodological Techniques: Bernstein Polynomial Approximation, Stieltjes Transform, and Martingale Decomposition

Proving the central limit theorem for LSS in the MP setting with a variance parameter involves several methodological innovations (Bai et al., 2010):

• Polynomial Approximation: Bernstein polynomial approximation is used to reduce sufficiently smooth (but not analytic) test functions to analytic approximants, enabling the use of contour integration and complex analysis tools in the proof.
• Truncation and Renormalization: The entries of $X_n$ are truncated and normalized to prevent heavy-tailed effects from violating moment assumptions.
• Stieltjes Transform and Martingale Expansion: The differences between the empirical and population spectral distributions are analyzed via their Stieltjes transforms. The fluctuations are decomposed using martingale difference techniques, with the quadratic forms and resolvent expansions finely analyzed to capture their joint limit.
• Contour Integration: The difference in linear statistics is represented as a contour integral involving the analytic approximants, leading to explicit mean and covariance formulas for the LSS process.

These techniques collectively allow the extension of previous CLTs (valid only for analytic test functions) to functions in $C^4$, and expose the roles of the variance parameter, entry kurtosis, and spectral structure.

5. Rate of Convergence, Local Laws, and High-Dimensional Fluctuations

Quantitative rates of convergence (measured, for instance, in the Kolmogorov distance) are controlled by the variance parameter, the entry moments, and the tail decay. For the standard i.i.d. model with $E[X_{jk}] = 0$, $E[X_{jk}^2] = 1$, and sub-exponential (or bounded fourth) moments,

$\sup_x |F_n(x) - F_y(x)| = O(n^{-1} \log^A n)$

with high probability, where constants depend on $y$ and moment parameters (Götze et al., 2011, Götze et al., 2014).

Furthermore, at the microscopic scale, local laws (describing the behavior in spectral windows containing only a few eigenvalues) also critically depend on the variance parameter. Near the hard edge $x \downarrow 0$, the density diverges as $x^{-1/2}$, and the local spacing is of order $\sqrt{x}/n$; controlling the accuracy of empirical density estimates at this scale requires a careful analysis of the variance normalization and Stieltjes transform (Cacciapuoti et al., 2012, Kafetzopoulos et al., 2022, Ajanki et al., 2013).

6. Extensions: Arbitrary Variance Profiles, Time Series, and Non-Standard Models

The notion of a variance parameter extends naturally to models with a non-constant variance profile. For random Gram matrices or covariance matrices with non-uniform variances $s_{ik}$, the limiting density is governed by a system of nonlinear self-consistent equations (Dyson or quadratic vector equations) in which $S = (s_{ik})$ replaces $y$ as the parameter controlling spectral features. This accommodates settings with block-dependent or heavy-tailed structures, as well as cases with high or low sparsity (Alt et al., 2016, Bryson et al., 2019, Castillo, 2022).

The MP law with an effective variance parameter also arises in functional CLTs for time series models with temporal dependence, expressed through frequency-dependent transfer functions $h(\lambda,\nu)$ (Liu et al., 2013). The limiting spectral distribution then depends on both cross-sectional and frequency variance, further generalizing the role of the variance parameter.

In the heavy-tailed regime (infinite variance), the empirical spectral distribution deviates from the classical MP law, but its low-order moments still match those of the MP law, with heavy-tail corrections explicitly identified as additive (starting from the fourth moment) (Heiny et al., 2020). This shows the robustness of the variance-parameter viewpoint but also highlights its limits in highly non-Gaussian settings.

7. Applications and Significance

The Marčenko–Pastur law with variance parameter underpins a wide range of high-dimensional statistical problems, including:

• Principal Component Analysis (PCA): The variance parameter $y$ determines the bulk spectrum and informs the identification of outlier (signal) eigenvalues.
• Hypothesis Testing and Estimation: CLTs for LSS, with explicit dependence on $y$, provide asymptotic distributions for spectral statistics widely used in covariance estimation and testing.
• Signal Processing and Wireless Communications: Spectral properties of Wishart-type matrices, with variance parameter reflecting system load, are fundamental in capacity calculations and code design.

In advanced scenarios, the explicit variance profile, higher-moment parameters, and extensions to tensors, block structures, and dependent settings ensure that the Marčenko–Pastur paradigm continues to serve as a foundational tool in both theoretical and applied research.

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Summary Table: Explicit Appearance of the Variance Parameter

Context	Appearance of Variance Parameter	Relevant Reference
Classical (i.i.d.) MP law	$y = p/n$; support $[(1-\sqrt{y})^2, (1+\sqrt{y})^2]$	(Bai et al., 2010, Götze et al., 2011)
Fluctuations/LSS CLT	$y$ in mean/covariance; higher moments via $\kappa_1, \kappa_2$	(Bai et al., 2010)
Arbitrary variance profile	$S = (s_{ik})$ in self-consistent equations	(Alt et al., 2016, Ajanki et al., 2013)
Local laws (hard edge)	Normalization and density singularity via variance	(Cacciapuoti et al., 2012, Kafetzopoulos et al., 2022)
High-dimensional time series	Frequency/spatial-dependent $h(\lambda,\nu)$	(Liu et al., 2013)
Block/tensor/structured models	Effective variance per block or per tensor index	(Bryson et al., 2019, Yaskov, 2021, Collins et al., 2021)

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The Marčenko–Pastur law with variance parameter thus encapsulates the interplay between high-dimensional geometry, variance normalization, spectral fluctuations, and model-specific structure, providing a precise and flexible framework for the paper of large random matrices and their spectral statistics.