Some estimates of norms of random matrices

Latała, Rafał

doi:10.1090/s0002-9939-04-07800-1

Cited by 184 publications

(171 citation statements)

References 7 publications

(6 reference statements)

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“…Surprisingly, however, no counterexamples to this statement are known for structured random matrices with independent Gaussian entries. This observation has led to the following conjecture proposed by R. Lata la (see also [4,5]). …”

Section: Introductionmentioning

confidence: 89%

“…The result of Theorem 1.4 is complementary to Theorem 1.1: while Theorem 1.1 is often sharp, Theorem 1.4 can give a substantial improvement for highly inhomogeneous matrices. For example, Theorem 1.4 readily implies the dimension-free bound of Lata la [4], which could not be reproduced using Theorem 1.1. The statement of Theorem 1.4 was chosen for sake of illustration; it is in fact a direct consequence of a sharper bound that arises from the proof.…”

Section: Theorem 14 the Expected Spectral Norm Satisfiesmentioning

confidence: 99%

“…To illustrate this, let us use Corollary 4.2 to derive a delicate (but much less sharp) result of Lata la [4] that could not be recovered from Theorem 1.1.…”

Section: Corollary 42mentioning

confidence: 99%

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On the spectral norm of Gaussian random matrices

Handel

2017

Trans. Amer. Math. Soc.

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Abstract. Let X be a d × d symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata la that the spectral norm of X is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order √ log log d. Moreover, dimensionfree bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes.

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Section: Introductionmentioning

confidence: 89%

Section: Theorem 14 the Expected Spectral Norm Satisfiesmentioning

confidence: 99%

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On the spectral norm of Gaussian random matrices

Handel

2017

Trans. Amer. Math. Soc.

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“…In the rest of the paper, we only consider asymptotics in which N and T grow at the same rate; that is, we could equivalently write Geman (1980), Silverstein (1989), Bai, Silverstein, and Yin (1988), Krishnaiah (1988), andLatala (2005). Loosely speaking, we expect the result e = O p ( max(N, T )) to hold as long as the errors e it have mean zero, uniformly bounded fourth moment, and weak time-serial and cross-sectional correlation (in some well-defined sense, see the examples).…”

Section: Expansion Of the Profile Objective Functionmentioning

confidence: 99%

“…(4) Assumption 5 is also sufficient for Assumption 3 * (and thus for Assumption 3), because e it is assumed independent over t and across i and has a bounded fourth moment, conditional on C, which by using results in Latala (2005), implies the spectral norm satisfies e = max(N, T ) as N and T become large; see the supplementary material.…”

Section: Remarks On Assumptionmentioning

confidence: 99%

Dynamic linear panel regression models with interactive fixed effects

Moon¹,

Weidner²

2014

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We analyze linear panel regression models with interactive fixed effects and predetermined regressors, for example lagged-dependent variables. The first-order asymptotic theory of the least squares (LS) estimator of the regression coefficients is worked out in the limit where both the cross-sectional dimension and the number of time periods become large. We find two sources of asymptotic bias of the LS estimator: bias due to correlation or heteroscedasticity of the idiosyncratic error term, and bias due to predetermined (as opposed to strictly exogenous) regressors. We provide a bias-corrected LS estimator. We also present bias-corrected versions of the three classical test statistics (Wald, LR, and LM test) and show their asymptotic distribution is a χ 2 -distribution. Monte Carlo simulations show the bias correction of the LS estimator and of the test statistics also work well for finite sample sizes.

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Invertibility of symmetric random matrices

Vershynin

2012

Random Struct Algorithms

114

313

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We study n × n symmetric random matrices H, possibly discrete, with iid above-diagonal entries. We show that H is singular with probability at most exp(−n c ), and. Furthermore, the spectrum of H is delocalized on the optimal scale o(n −1/2 ). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdös, Schlein and Yau.

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Some estimates of norms of random matrices

Abstract: Abstract. We show that for any random matrix (X ij ) with independent mean zero entries

Cited by 184 publications

References 7 publications

On the spectral norm of Gaussian random matrices

On the spectral norm of Gaussian random matrices

Dynamic linear panel regression models with interactive fixed effects

Invertibility of symmetric random matrices

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