This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in [Tro11c] that yields both upper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices. This machinery is used to derive eigenvalue analogs of the classical Chernoff, Bennett, and Bernstein bounds. Two examples demonstrate the efficacy of the minimax Laplace transform. The first concerns the effects of column sparsification on the spectrum of a matrix with orthonormal rows. Here, the behavior of the singular values can be described in terms of coherence-like quantities. The second example addresses the question of relative accuracy in the estimation of eigenvalues of the covariance matrix of a random process. Standard results on the convergence of sample covariance matrices provide bounds on the number of samples needed to obtain relative accuracy in the spectral norm, but these results only guarantee relative accuracy in the estimate of the maximum eigenvalue. The minimax Laplace transform argument establishes that if the lowest eigenvalues decay sufficiently fast, Ω(e^(-2)κ^2_l l log p) samples, where κ_l = λ_1(C) λ_l(C), are sufficient to ensure that the dominant l eigenvalues of the covariance matrix of a N(0,C) random vector are estimated to within a factor of 1 ± e with high probability.
The modern asymptotic theory began in the 1950s when physicists observed that, on certain scales, the behavior of a quantum system is described by the spectrum of a random matrix @cite_49 . They further observed the phenomenon of : as the dimension increases, the spectral statistics become independent of the distribution of the random matrix; instead, they are determined by the symmetries of the distribution @cite_50 . Since these initial observations, physicists, statisticians, engineers, and mathematicians have found manifold applications of the asymptotic theory in high-dimensional statistics @cite_20 @cite_9 @cite_21 , physics @cite_16 @cite_49 , wireless communication @cite_4 @cite_75 , and pure mathematics @cite_72 @cite_14 , to mention only a few areas.
@cite_14 @cite_4 @cite_9 @cite_21 @cite_72 @cite_50 @cite_49 @cite_16 @cite_75 @cite_20
2108600031 2044172468 2002700608 2125319092
In the last few years, the asymptotic distribution of the singular values of certain random matrices has emerged as a key tool in the analysis and design of wireless communication channels. These channels are characterized by random matrices that admit various statistical descriptions depending on the actual application. The goal of this paper is the investigation and application of random matrix theory with particular emphasis on the asymptotic theorems on the distribution of the squared singular values under various assumption on the joint distribution of the random matrix entries. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino, Caire, Shamai and Verdu. We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1 2 and nott1 2 as in the classical case. We find the covariance matrix of the limit distribution. The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries. We focus on recently developed geometric methods for estimating the hard edge of random matrices (the smallest singular value).