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 literature on the existence and universality of LSDs is massive; we mention only the highlights. It is now known that the semicircular law is universal for Wigner matrices. Suppose that @math is a sequence of independent @math Wigner matrices. Grenander established that if all the moments are finite, then the ESD of @math converges weakly to the semicircular law in probability @cite_73 . Arnold showed that, assuming a finite fourth moment, the ESD almost surely converges weakly to the semicircular law @cite_40 . Around the same time, Mar c enko and Pastur determined the form of the limiting spectral distribution of sample covariance matrices @cite_25 . More recently, Tao and Vu confirmed the long-conjectured circular law hypothesis. Let @math be a sequence of independent @math matrices whose entries are i.i.d. and have unit variance. Then the ESD of @math converges weakly to the uniform measure on the unit disk, both in probability and almost surely @cite_17 .
@cite_40 @cite_73 @cite_25 @cite_17
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A process and an apparatus for the continuous ozonization of unsaturated organic compounds in the presence of water, wherein (a) the fresh charge mixture of unsaturated compounds to be ozonized is reacted in at least one first reactor in parallel flow with ozonic gas, which was previously used for ozonizing a previously partially ozonized charge mixture, and (b) simultaneously a previously partially ozonized charge mixture is reacted in at least one second reactor in parallel flow with fresh ozonic gas, and wherein, optionally, said at least two reactors contain at least two mixing sections for mixing the charged liquid and gaseous phases with each other where the hydraulic diameter of the mixing elements in the individual mixing section is reduced in the flow direction. In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space v, a selfadjoint operator BN (re) of the form Given an n x n complex matrix A, let mu(A)(x, y) := 1 n vertical bar 1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1 root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1 root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1 root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.