Some Remarks on the Dozier–silverstein Theorem for Random Matrices With Dependent Entries

Random Struct Algorithms

Wolff

2015

Self Cite

Abstract. An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on Hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest.

“…Indeed, one can construct

{boldx}^{(n)}

in (2), satisfying (A1‐A2), in such a way that

K_{n} = O (n^{1 / 2})

while 0 is an eigenvalue of

X^{(n)}

of multiplicity at least cn , which would contradict the fact that ν z is absolutely continuous. Proof of Theorem First of all, note that since

K_{n} = o (n^{1 / 2})

, we have

E | X_{11}^{(n)} |^{4} = o (n) and E | X_{11}^{(n)} |^{3} = o (n^{1 / 2}) .

K_{n} = O (1)

, the result follows from , Theorem 2.11 and Proposition 2.12]. In fact one can check that the proof given there works for K n being a small power of n .…”

Section: Limiting Distribution Of the Singular Values Of Shiftsmentioning

confidence: 84%

Section: Discussion and Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Circular law for random matrices with exchangeable entries

Adamczak

Random Struct Algorithms

Wolff

2015

Self Cite

“…rows, such as for certain random Markov matrices [11], for random matrices with i.i.d. log-concave rows [1,2], for random ±1 matrices with i.i.d. rows of given sum [35] (see also [40]), etc.…”

Section: Introductionmentioning

confidence: 99%

“…This is in contrast with the model of random matrices with i.i.d. log-concave rows studied in [1], for which it turned out that the circular law holds without assuming unconditionality, as explained in [2].…”

Section: Introductionmentioning

confidence: 99%

Circular law for random matrices with unconditional log-concave distribution

Adamczak

2015

Commun. Contemp. Math.

Self Cite

Abstract. We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n × n real matrix having as a random vector in R n×n a log-concave isotropic unconditional law. In particular, the entries are uncorellated and have a symmetric law of zero mean and unit variance. This allows for some dependence and non equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries. Our main result states that as n goes to infinity, the empirical spectral distribution of 1 √ n A tends to the uniform law on the unit disc of the complex plane.

On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Probab. Theory Relat. Fields

Tikhomirov

2017

International audienceConsider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries