Limit Theorem For Linear Statistics Research Articles

Gaussian limit for determinantal point processes with J-Hermitian kernels

We show that the central limit theorem for linear statistics over determinantal point processes with J-Hermitian kernels holds under fairly general conditions. In particular, we establish the Gaussian limit for linear statistics over determinantal point processes on the union of two copies of ℝd when the correlation kernels are J-Hermitian translation-invariant.

On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation

In this paper, we propose a new approach to the central limit theorem (CLT) based on functions of bounded Fréchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g. the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, [Formula: see text]. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in [Formula: see text].

Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others.Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker–Campbell–Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero.We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications

We give a simple proof of a central limit theorem for linear statistics of the circular $\beta $-ensembles which is valid at almost microscopic scales for functions of class $C^{3}$. Using a coupling introduced by Valko and Virag [48], we deduce a central limit theorem for the Sine$_{\beta }$ processes. We also discuss connections between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of [37], we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the circular $\beta $-ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This establishes that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions coming from log-correlated Gaussian field theory.

Fixed energy universality of Dyson Brownian motion

We consider Dyson Brownian motion for classical values of β with deterministic initial data V. We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time t≳1/N if the density of states of V is bounded above and below down to scales η≪t in a window of size L≫t. Our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known. Our methodology builds on the homogenization theory developed in [16] which reduces the microscopic problem to a mesoscopic problem. As an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical Dyson Brownian motion.

Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general $$\beta $$ β and potentials

We study Dyson Brownian motion with general potential V and for general $$\beta \ge 1$$ . For short times $$t = o (1)$$ and under suitable conditions on V we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $$\eta _* \ll t \ll 1$$ , we prove a mesoscopic central limit theorem for linear statistics at all scales $$\eta $$ with $$N^{-1}\ll \eta \ll t$$ .

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of $$\alpha $$ -mixing (for local statistics) and exponential $$\alpha $$ -mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.

On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

Summary By studying the family of p-dimensional scale mixtures, the paper shows for the first time a non-trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated Marčenko–Pastur law. A different and new limit is found and characterized. The reasons for failure of the Marčenko–Pastur limit in this situation are found to be a strong dependence between the p-co-ordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analyse the traditional John's-type test we establish a novel and general central limit theorem for linear statistics of eigenvalues of the sample covariance matrix. It is shown that John's test and its recent high dimensional extensions both fail for high dimensional mixtures, precisely because of the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used integrated classification likelihood and Bayesian information criteria in detecting non-spherical component covariance matrices of a high dimensional mixture.

Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function

We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The result mirrors central limit theorems in random matrix theory that have been proved by Szegő, Spohn, and Soshnikov among others, and therefore provides support for the view that the zeros of the zeta function are distributed like the eigenvalues of a random matrix. A key ingredient in our proof is a simple bootstrapping of classical zero density estimates of Selberg and Jutila for the zeta function, which may be of independent interest.

Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.

CENTRAL LIMIT THEOREM FOR LINEAR STATISTICS OF EIGENVALUES OF BAND RANDOM MATRICES

We prove the Central Limit Theorem for linear statistics of the eigenvalues of band random matrices provided [Formula: see text] and test functions are sufficiently smooth.

Fluctuations of deformed Wigner random matrices

Let Xn be a standard real symmetric (complex Hermitian) Wigner matrix, y1, y2, ..., yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn, α = n−1/2Xn + n−α/2(y1,..., yn), where 0 < α < 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn, α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation \(\mathbb{E}m_{n,\alpha } \left( z \right)\) and variance Var(mn, α(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation.

Fluctuations of eigenvalues for random Toeplitz and related matrices

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,\cdots,$ being independent real random variables such that $$ \mathbb{E}[a_{j}]=0, \mathbb{E} [|a_{j}|^{2}]=1 \mathrm{for}\, j=0,1,2,\cdots,$$ (homogeneity of 4-th moments) $$\kappa=\mathbb{E} [|a_{j}|^{4}],$$ and further (uniform boundedness) $$\sup\limits_{j\geq 0} \mathbb{E} [|a_{j}|^{k}]=C_{k}<\infty \mathrm{for} k\geq 3.$$ Under the assumption of $a_{0}\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed polynomial with degree at least 2. Without this assumption, the CLT can be easily modified to a possibly non-normal limit law. In a special case where $a_{j}$'s are Gaussian, the result has been obtained by Chatterjee for some test functions. Our derivation is based on a simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our method can apply to other related random matrix models, including Hermitian Toeplitz and symmetric Hankel matrices. Since Toeplitz matrices are quite different from Wigner and Wishart matrices, our results enrich this topic.

Rates of convergence in the central limit theorem for linear statistics of martingale differences

In this paper, we give rates of convergence for minimal distances between linear statistics of martingale differences and the limiting Gaussian distribution. In particular the results apply to the partial sums of (possibly long range dependent) linear processes, and to the least squares estimator in some parametric regression models.

Gaussian fluctuations for non-Hermitian random matrix ensembles

Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f)=∑k=1Nf(λk) where λ1, λ2, …, λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.

Functional limit theorems for linear statistics from sequential ranks

Let X 1 ,..., X n be a sequence of continuously distributed independent random variables. The normalized ranks R kn and sequential ranks S k , k=1,...,n, are defined by $${\text{R}}_{{\text{kn}}} = \frac{1}{{\text{n}}}\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} ,} {\text{ S}}_{\text{k}} = \frac{1}{{\text{k}}}\sum\limits_{{\text{j = }}1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} .} $$ The subject of the present paper is the asymptotic behavior, as n→∞, of the process $$\frac{1}{{\sqrt {\text{n}} }}\sum\limits_{{\text{k}} \leqq {\text{nt}}} {{\text{a}}({\text{S}}_{\text{k}} ),} {\text{ }}0 \leqq {\text{t}} \leqq 1,$$ for a∈L 2 (0, 1), $$\int\limits_0^1 {{\text{adn}} = 0} $$ . For suitable a, the limiting law of that process is expressed as solution of a stochastic equation under the hypothesis of identically distributed X 1,..., X n as well as under a class of contiguous alternatives, which contains the occurrence of a change point in the series of measurements.