Central Limit Theorems for Linear Spectral Statistics

• The paper establishes a rigorous derivation of Gaussian fluctuations in linear spectral statistics using contour integrals and explicit kernel formulas.
• It extends the CLT framework to various covariance structures including spiked, separable, and mesoscopic regimes, enhancing high-dimensional inference.
• Practical computation relies on Silverstein fixed-point methods and quadrature rules to accurately evaluate limiting spectral distributions.

Central limit theorems (CLTs) for linear spectral statistics (LSS) are fundamental results in high-dimensional random matrix theory, characterizing the global fluctuations of eigenvalue statistics of large random matrices, notably sample covariance and related structured ensembles. These CLTs rigorously specify the Gaussian fluctuations—explicit mean and covariance—of quantities of the form $\sum_{i=1}^p f(\lambda_i)$, where $f$ is a (suitably regular) test function and $\lambda_i$ are the eigenvalues of the random matrix of interest. The scope of LSS CLTs covers classical sample covariance matrices, their generalizations with dependent or non-Gaussian entries, spiked or deformed models, separable structures, high-dimensional correlation matrices, as well as models arising in random graphs and block-structured ensembles. These results underpin numerous developments in multivariate statistics, statistical signal processing, high-dimensional inference, and hypothesis testing.

1. Universal Structure of CLT for Linear Spectral Statistics

The prototypical setting considers a $p \times n$ data matrix $X = (x_{ij})$ with independent entries, centralized to mean zero and appropriately scaled (typically $E[x_{ij}]=0$, $E[|x_{ij}|^2]=1$), and forms the sample covariance matrix

$B_n = \frac1n Q X X^* Q^*,$

where $Q$ is a deterministic, possibly rectangular, or even infinite-dimensional $p \times k$ mixing matrix. The empirical spectral distribution (ESD) of $B_n$ converges to the generalized Marčenko–Pastur law $F^{y,H}$, with $y = \lim p/n$ and $H$ the limiting spectral distribution of the “population” covariance $T_n = Q Q^*$. For any analytic function $f$,

$$S_n(f) = \sum_{i=1}^p f(\lambda_i) = p \int f(x)\,dF^{B_n}(x)$$

satisfies

$$S_n(f) - p F^{y,H}(f) \xrightarrow{d} N\big(\mu(f), \sigma^2(f)\big),$$

where both $\mu(f)$ and $\sigma^2(f)$ admit explicit contour-integral expressions in terms of the test function $f$ and the limiting Stieltjes transform $\underline m(z)$ determined by the Silverstein fixed-point equations (Zheng et al., 2017).

The mean and covariance are typically given by

$$\begin{aligned} \mu(f) & = -\frac{1}{2\pi i} \oint f(z)\left\{ \alpha_x \frac{A(z)}{B(z)} + \beta_x \frac{C(z)}{D(z)} \right\} dz, \ \sigma^2(f) & = C_1(f) + C_2(f) + C_3(f), \end{aligned}$$

with $C_1(f)$ the “universal” variance kernel, $C_2(f)$ and $C_3(f)$ encoding effects due to the excess kurtosis (fourth cumulant) and possible complexification. The general structure persists across a very broad class of models; the explicit expressions for $A(z)$, $B(z)$, $C(z)$, $D(z)$, and the kernels depend on the population spectrum and the specifics of the random matrix ensemble.

2. Generalizations: Dependent Data, Infinite Mixing, and Separable Covariance

The foundational CLT of Bai–Silverstein (2004) has been extended to highly general settings, including mixing structures (e.g., $Q$ permitting $k > p$ or $k = \infty$), separable covariance, and complex “block” models.

• Dependent Data and Infinite-Dimensional Mixing: For $k$ possibly infinite, $Q$ can encode, for instance, mixing for repeated linear processes such as ARMA, allowing the model to encompass high-dimensional time series with complex temporal or spatial dependencies. The limiting spectral law remains governed by a generalized Marčenko–Pastur equation, and the LSS CLT is preserved, with technical modifications for the control of higher moments and population spectrum regularity (Zheng et al., 2017).
• Separable Sample Covariance and General Variance Profiles: Separable models,

$$B_n = \frac{1}{N} T_{2n}^{1/2}\, X_n\, T_{1n}\, X_n'\, T_{2n}^{1/2},$$

arise naturally in settings with two-way structured covariance (e.g., spatio-temporal or Kronecker-product models). The LSS CLT holds with explicit expressions for mean and covariance as multidimensional contour integrals, involving implicitly defined functions $g_1(z), g_2(z), m(z)$ that solve a system determined by the spectral distributions of $T_{1n}$ and $T_{2n}$ (Zhidong et al., 2016, Li et al., 2019).

• Block, Deformed, and Graphon Models: CLTs for ensembles with block-wise variance and dependencies (such as stochastic block models) or inhomogeneous random graphs are established via explicit formulas involving multi-block Stieltjes transforms, mean/covariance kernels, and graphon homomorphism densities (Zhu et al., 2024, Wang et al., 2021).

3. High-Dimensional and Non-Gaussian Extensions

The regime $p, n \rightarrow \infty$ with their ratio tending to a constant (including $p > n$ or $p \gg n$) is fully covered, with several subtleties:

• $p/n \gg 1$ “Ultra-High-Dimensional” Limit: The LSS CLT adapts with the limiting spectral law changing from Marčenko–Pastur to the semicircle law. New correction terms in the mean arise, and the covariance kernel includes a $\nu_4 - 3$ factor (where $\nu_4$ is the fourth moment), reflecting the universality-breaking role of kurtosis in the ultra-high-dimensional regime (Chen et al., 2015).
• Non-Gaussianity, Heavy Tails, and Sample Correlation: The LSS CLT holds under general moment or Lindeberg-type conditions. For sample correlation matrices—or other self-normalized statistics—a necessary and sufficient condition for the universal CLT is $$\lim_{x\rightarrow\infty} x^3\mathbb{P}(|\xi|>x) = 0$$, corresponding to finite $3+\varepsilon$ moments. When $\alpha < 3$ in tails $\mathbb{P}(|\xi|>x)\sim l(x)x^{-\alpha}$, the variance scales differently and the universal CLT breaks; otherwise, the mean and variance match the finite fourth-moment case (Li et al., 2024).
• Rank Correlation (Kendall’s $\tau$): For robust, nonparametric Wigner-like random matrices, all LSS admit a universal CLT with deterministic mean shift and covariance given by modified resolvent formulas, with no moment assumptions required beyond continuity (Li et al., 2019).

4. Spiked and Hypothesis Testing Applications

Spiked models and hypothesis testing in high dimensions rely fundamentally on the LSS CLT structure.

• Spiked Models: In both “classical” and generalized spike settings, the LSS CLT introduces an $O(1)$ additive correction due to a finite number of outlying (spiked) eigenvalues; the variance remains the same as in the unspiked case. Contour-integral and Coulomb fluid methods yield explicit formulas for these corrections (Passemier et al., 2014, Passemier et al., 2014, Liu et al., 5 Oct 2025).
• Testing: Linear spectral statistics yield optimal or near-optimal test statistics for equality of covariance (sphericity), white noise in time series, and spiked alternatives. For example, the sphericity test $f(x) = (x-1)^2$ yields a Gaussian limit with mean and variance computable via the CLT, enabling explicit critical value calculations in high dimensions. Analogous results hold for Wilks’ $\log\det$, Lawley–Hotelling, Bartlett–Nanda–Pillai, and Roy’s largest root tests, with full power expressions under rank-$M$ alternatives deriving from the generalized spiked LSS CLT (Zheng et al., 2017, Li et al., 2018, Liu et al., 5 Oct 2025).
• Joint CLTs and White-Noise Testing: The joint LSS CLT for several dependent sample covariance matrices enables multivariate hypothesis testing for time series and functional data, where cross-covariances are captured through joint spectral limits, extending beyond mere marginal calculations (Li et al., 2018).

5. Mesoscopic and Local Spectral Fluctuations

Developments in mesoscopic scaling situate the LSS CLT in a multiscale context:

• Mesoscopic CLTs: For test functions localized on a window of width $\eta \gg N^{-1}$ but $\eta \ll 1$, the mesoscopic LSS CLT ensures convergence to a universal Gaussian process, with explicitly computable variance kernel (Fourier transform of $|\xi|\cdot |\widehat f(\xi)|^2$). At spectral edges, the mean may acquire an $O(1)$ shift (Li et al., 2019).
• High-Dimensional Local Statistics: In the ultra-high-dimensional regime, the local LSS CLT reveals universality at the mesoscopic scale: the impact of non-Gaussianity (fourth cumulant) disappears, reflected in the independence of the covariance kernel from higher moments, in contrast to the global CLT (Ding et al., 2023).

6. Functional and Multi-Function Extensions

The functional CLT extends LSS fluctuations to the space of all admissible test functions, equipping the set of LSSs with a Gaussian process structure whose mean and covariance kernels are given by double-integral or functional expressions involving the Stieltjes transform and its derivatives. Bernstein polynomial approximation and martingale expansions are standard proof devices, and the limiting random process admits explicit covariance and mean kernels expressed in terms of contour integrals or variational representations (Bai et al., 2010).

7. Implementation, Computation, and Practicalities

All main CLT formulas are given in terms of Stieltjes transforms and population spectral distributions, enabling practical numerical evaluation. The Silverstein equations, which underlie most models, can be solved via standard fixed-point iteration or Newton methods, and the contour integrals can be evaluated by quadrature, e.g., trapezoidal rules on circles in the complex plane, with error estimates decaying exponentially in the number of points used (Zheng et al., 2017, Xie et al., 2024). Adaptation to finite-sample, non-centralized, or other contrived statistics generally requires only the appropriate scaling or substitution of sample size parameters, as established in the comparison of centralized versus non-centralized covariance matrices (Zheng et al., 2013).

---

In summary, CLTs for linear spectral statistics constitute a comprehensive framework for understanding global eigenvalue fluctuations in random matrix ensembles. The unified theoretical machinery applies—sometimes with minimal adaptation—to a wide range of random matrix models: from classical to generalized, dense to sparse, real to complex, independent to dependent, with comparisons and joint fluctuations, and for a broad class of test functions. These results underpin high-dimensional inference and provide the technical foundation for optimal hypothesis testing and multivariate statistics in modern data science (Zheng et al., 2017, Zhidong et al., 2016, Li et al., 2024, Zhu et al., 2024, Liu et al., 5 Oct 2025).