Displaying similar documents to “On bilinear forms based on the resolvent of large random matrices”

Comparison between two types of large sample covariance matrices

Guangming Pan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let { X i j } , i , j = , be a double array of independent and identically distributed (i.i.d.) real random variables with E X 11 = μ , E | X 11 - μ | 2 = 1 and E | X 11 | 4 l t ; . Consider sample covariance matrices (with/without empirical centering) 𝒮 = 1 n j = 1 n ( 𝐬 j - 𝐬 ¯ ) ( 𝐬 j - 𝐬 ¯ ) T and 𝐒 = 1 n j = 1 n 𝐬 j 𝐬 j T , where 𝐬 ¯ = 1 n j = 1 n 𝐬 j and 𝐬 j = 𝐓 n 1 / 2 ( X 1 j , ... , X p j ) T with ( 𝐓 n 1 / 2 ) 2 = 𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n with p / n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the...

( 0 , 1 ) -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let m and n be positive integers, and let R = ( r 1 , ... , r m ) and S = ( s 1 , ... , s n ) be nonnegative integral vectors. Let A ( R , S ) be the set of all m × n ( 0 , 1 ) -matrices with row sum vector R and column vector S . Let R and S be nonincreasing, and let F ( R ) be the m × n ( 0 , 1 ) -matrix, where for each i , the i th row of F ( R , S ) consists of r i 1’s followed by ( n - r i ) 0’s. Let A A ( R , S ) . The discrepancy of A, disc ( A ) , is the number of positions in which F ( R ) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over...

Persistence of iterated partial sums

Amir Dembo, Jian Ding, Fuchang Gao (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let S n ( 2 ) denote the iterated partial sums. That is, S n ( 2 ) = S 1 + S 2 + + S n , where S i = X 1 + X 2 + + X i . Assuming X 1 , X 2 , ... , X n are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities p n ( 2 ) : = max 1 i n S i ( 2 ) l t ; 0 c 𝔼 | S n + 1 | ( n + 1 ) 𝔼 | X 1 | , with c 6 30 (and c = 2 whenever X 1 is symmetric). The converse inequality holds whenever the non-zero min ( - X 1 , 0 ) is bounded or when it has only finite third moment and in addition X 1 is squared integrable. Furthermore, p n ( 2 ) n - 1 / 4 for any non-degenerate squared integrable, i.i.d., zero-mean X i . In contrast, we show that for any 0 l t ; γ l t ; 1 / 4 there exist integrable,...

Uniform mixing time for random walk on lamplighter graphs

Júlia Komjáthy, Jason Miller, Yuval Peres (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Suppose that 𝒢 is a finite, connected graph and X is a lazy random walk on 𝒢 . The lamplighter chain X associated with X is the random walk on the wreath product 𝒢 = 𝐙 2 𝒢 , the graph whose vertices consist of pairs ( f ̲ , x ) where f is a labeling of the vertices of 𝒢 by elements of 𝐙 2 = { 0 , 1 } and x is a vertex in 𝒢 . There is an edge between ( f ̲ , x ) and ( g ̲ , y ) in 𝒢 if and only if x is adjacent to y in 𝒢 and f z = g z for all z x , y . In each step, X moves from a configuration ( f ̲ , x ) by updating x to y using the transition rule of X and then...

Computing the greatest 𝐗 -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector x is said to be an eigenvector of a square max-min matrix A if A x = x . An eigenvector x of A is called the greatest 𝐗 -eigenvector of A if x 𝐗 = { x ; x ̲ x x ¯ } and y x for each eigenvector y 𝐗 . A max-min matrix A is called strongly 𝐗 -robust if the orbit x , A x , A 2 x , reaches the greatest 𝐗 -eigenvector with any starting vector of 𝐗 . We suggest an O ( n 3 ) algorithm for computing the greatest 𝐗 -eigenvector of A and study the strong 𝐗 -robustness. The necessary and sufficient conditions for strong 𝐗 -robustness are introduced...

On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let 𝕄 n , m be the set of all n × m real or complex matrices. For A , B 𝕄 n , m , we say that A is row-sum majorized by B (written as A rs B ) if R ( A ) R ( B ) , where R ( A ) is the row sum vector of A and is the classical majorization on n . In the present paper, the structure of all linear operators T : 𝕄 n , m 𝕄 n , m preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on n and then find the linear preservers of row-sum majorization of these relations on 𝕄 n , m . ...

Distance matrices perturbed by Laplacians

Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel (2020)

Applications of Mathematics

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Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s . Let D i j denote the sum of all the weights lying in the path connecting the vertices i and j of T . We now say that D i j is the distance between i and j . Define D : = [ D i j ] , where D i i is the s × s null matrix and for i j , D i j is the distance between i and j . Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of order...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let A , B be n × m matrices. The concept of matrix majorization means the j th column of A is majorized by the j th column of B and this is done for all j by a doubly stochastic matrix D . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

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If the space 𝒬 of quadratic forms in n is splitted in a direct sum 𝒬 1 ... 𝒬 k and if X and Y are independent random variables of n , assume that there exist a real number a such that E ( X | X + Y ) = a ( X + Y ) and real distinct numbers b 1 , . . . , b k such that E ( q ( X ) | X + Y ) = b i q ( X + Y ) for any q in 𝒬 i . We prove that this happens only when k = 2 , when n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

Gaussian approximation of Gaussian scale mixtures

Gérard Letac, Hélène Massam (2020)

Kybernetika

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For a given positive random variable V > 0 and a given Z N ( 0 , 1 ) independent of V , we compute the scalar t 0 such that the distance in the L 2 ( ) sense between Z V 1 / 2 and Z t 0 is minimal. We also consider the same problem in several dimensions when V is a random positive definite matrix.

On the combinatorial structure of 0 / 1 -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A 0 / 1 -simplex is the convex hull of n + 1 affinely independent vertices of the unit n -cube I n . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0 / 1 -simplices in I n can be represented by 0 / 1 -matrices P of size n × n whose Gramians G = P P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse P - of P is doubly stochastic and has the...

On linear preservers of two-sided gut-majorization on 𝐌 n , m

Asma Ilkhanizadeh Manesh, Ahmad Mohammadhasani (2018)

Czechoslovak Mathematical Journal

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For X , Y 𝐌 n , m it is said that X is gut-majorized by Y , and we write X gut Y , if there exists an n -by- n upper triangular g-row stochastic matrix R such that X = R Y . Define the relation gut as follows. X gut Y if X is gut-majorized by Y and Y is gut-majorized by X . The (strong) linear preservers of gut on n and strong linear preservers of this relation on 𝐌 n , m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of gut on n and 𝐌 n , m .

Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper, we study the size of the giant component C G in the random geometric graph G = G ( n , r n , f ) of n nodes independently distributed each according to a certain density f ( · ) in [ 0 , 1 ] 2 satisfying inf x [ 0 , 1 ] 2 f ( x ) g t ; 0 . If c 1 n r n 2 c 2 log n n for some positive constants c 1 , c 2 and n r n 2 as n , we show that the giant component of G contains at least n - o ( n ) nodes with probability at least 1 - e - β n r n 2 for all n and for some positive constant β . We also obtain estimates on the diameter and number of the non-giant components of G .

Positivity of integrated random walks

Vladislav Vysotsky (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Take a centered random walk S n and consider the sequence of its partial sums A n : = i = 1 n S i . Suppose S 1 is in the domain of normal attraction of an α -stable law with 1 l t ; α 2 . Assuming that S 1 is either right-exponential (i.e. ( S 1 g t ; x | S 1 g t ; 0 ) = e - a x for some a g t ; 0 and all x g t ; 0 ) or right-continuous (skip free), we prove that { A 1 g t ; 0 , , A N g t ; 0 } C α N 1 / ( 2 α ) - 1 / 2 as N , where C α g t ; 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Row Hadamard majorization on 𝐌 m , n

Abbas Askarizadeh, Ali Armandnejad (2021)

Czechoslovak Mathematical Journal

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An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let 𝐌 m , n be the set of all m × n real matrices. For A , B 𝐌 m , n , we say that A is row Hadamard majorized by B (denoted by A R H B ) if there exists an m × n row stochastic matrix R such that A = R B , where X Y is the Hadamard product (entrywise product) of matrices X , Y 𝐌 m , n . In this paper, we consider the concept of row Hadamard majorization as a relation on 𝐌 m , n and characterize the structure of all linear operators T : 𝐌 m , n 𝐌 m , n preserving...