scholarly journals Small Ball Probability for the Condition Number of Random Matrices

2020 ◽  
pp. 125-137
Author(s):  
Alexander E. Litvak ◽  
Konstantin Tikhomirov ◽  
Nicole Tomczak-Jaegermann
Keyword(s):  
Random Matrices ◽  
Condition Number ◽  
Small Ball Probability ◽  
Small Ball

scholarly journals Small ball probability estimates in terms of width

2005 ◽  
Vol 169 (3) ◽  
pp. 305-314 ◽  
Author(s):  
Rafał Latała ◽  
Krzysztof Oleszkiewicz
Keyword(s):  
Small Ball Probability ◽  
Small Ball ◽  
Probability Estimates

scholarly journals On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Charles El-Nouty
Keyword(s):  
General Result ◽  
Gaussian Processes ◽  
Brownian Sheet ◽  
Random Vectors ◽  
Fractional Brownian Sheet ◽  
Integral Test ◽  
Small Ball Probability ◽  
Small Ball ◽  
Sufficiency Part ◽  
Necessity Part

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

scholarly journals Complex random matrices have no real eigenvalues

2018 ◽  
Vol 07 (01) ◽  
pp. 1750014 ◽  
Author(s):  
Kyle Luh
Keyword(s):  
Random Matrices ◽  
Random Variables ◽  
Principal Component ◽  
Random Variable ◽  
Singular Values ◽  
Singular Value ◽  
Small Ball ◽  
Circular Law ◽  
Least Singular Value ◽  
Tail Bound

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

scholarly journals Zero-free neighborhoods around the unit circle for Kac polynomials

2021 ◽  
Author(s):  
Gerardo Barrera ◽  
Paulo Manrique
Keyword(s):  
Unit Circle ◽  
Random Variables ◽  
Common Denominator ◽  
Small Ball Probability ◽  
Small Ball ◽  
Probability Inequalities ◽  
Independent And Identically Distributed

AbstractIn this paper, we study how the roots of the Kac polynomials $$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$ W n ( z ) = ∑ k = 0 n - 1 ξ k z k concentrate around the unit circle when the coefficients of $$W_n$$ W n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as $$n\rightarrow \infty $$ n → ∞ if and only if $${\mathbb {E}}[\log ( 1+ |\xi _0|)]<\infty $$ E [ log ( 1 + | ξ 0 | ) ] < ∞ . Under the condition $${\mathbb {E}}[\xi ^2_0]<\infty $$ E [ ξ 0 2 ] < ∞ , we show that there exists an annulus of width $${\text {O}}(n^{-2}(\log n)^{-3})$$ O ( n - 2 ( log n ) - 3 ) around the unit circle which is free of roots with probability $$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$ 1 - O ( ( log n ) - 1 / 2 ) . The proof relies on small ball probability inequalities and the least common denominator used in [17].

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