Related papers: Finite N Fluctuation Formulas for Random Matrices

The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities. Notwithstanding a long history, a formulation providing high precision…

We continue investigations of our previous papers, in which there were proved central limit theorems (CLT) for linear eigenvalue statistics Tr f(M_n) and there were found the limiting probability laws for the normalised matrix elements of…

A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to\infty$ version of this correspondence by matching…

Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. We are interested in these statistics because they…

Let $F_n$ denote the distribution function of the normalized sum $Z_n = (X_1 + \dots + X_n)/\sigma\sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal…

We consider the ensemble of curves $\{\gamma_{\alpha,N}:\alpha\in(0,1],N\in\N\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}\sum_{n=0}^{N'-1}\exp(\pi i n^2\alpha)$, $0\leq N'<N$. We prove the…

Given any fixed $N \times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are…

We study the probability distribution of the index ${\mathcal N}_+$, i.e., the number of positive eigenvalues of an $N\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large $\mathcal{N}_+$ with the fraction…

For certain types of statistical models, the characteristic function (Fourier transform) is available in closed form, whereas the probability density function has an intractable form, typically as an infinite sum of probability weighted…

We consider the fluctuation of linear eigenvalue statistics of random band $n\times n$ matrices whose entries have the form $\mathcal{M}_{ij}=b^{-1/2}u^{1/2}(|i-j|)\tilde w_{ij}$ with i.i.d. $w_{ij}$ possessing the $(4+\varepsilon)$th…

We consider the problem of estimating the density $\Pi$ of a determinantal process $N$ from the observation of $n$ independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish…

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as…

We consider $n\times n$ random matrices $M_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha }\otimes \mathbf{y}_{\alpha }$, where $\tau _{\alpha }\in \mathbb{R}$, $\{\mathbf{y}_{\alpha }\}_{\alpha =1}^{m}$ are i.i.d. isotropic…

We consider $N\times N$ Gaussian random matrices, whose average density of eigenvalues has the Wigner semi-circle form over $[-\sqrt{2},\sqrt{2}]$. For such matrices, using a Coulomb gas technique, we compute the large $N$ behavior of the…

In statistical problems, a set of parameterized probability distributions is used to estimate the true probability distribution. If Fisher information matrix at the true distribution is singular, then it has been left unknown what we can…

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By…

This paper studies the properties of the probability density function $p_{\alpha,\nu, n}(\mathbf{x})$ of the $n$-variate generalized Linnik distribution whose characteristic function $\varphi_{\alpha,\nu,n}(\boldsymbol{t})$ is given by…

We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\ge0)$ and its empirical spectral measure process $\mu^{(n)}=(\mu_{t}^{(n)};t\ge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an…

Products of random $2\times 2$ matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighborhood of order $\lambda>0$ of…

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and…