High-dimensional sample covariance matrices with Curie-Weiss entries

被引：5

作者：

Fleermann, Michael ^{[1
]}

Heiny, Johannes ^{[2
]}

机构：

[1] Fernuniv, Fak Math & Informat, Univ Str 1, D-58084 Hagen, Germany

[2] Ruhr Univ Bochum, Fak Math, Univ Str 150, D-44801 Bochum, Germany

来源：

ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS | 2020年 / 17卷 / 02期

关键词：

Curie-Weiss; random matrix; Marcenko-Pastur law; semicircle law; high dimension; dependent entries; full correlation; SEMICIRCLE LAW; EIGENVALUES; CONVERGENCE; INDEPENDENCE; LIMIT;

D O I：

10.30757/ALEA.v17-33

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

We study the limiting spectral distribution of sample covariance matrices XXT, where X are p x n random matrices with correlated entries and p/n -> y is an element of [0, infinity). If y > 0, we obtain the Marcenko-Pastur distribution and in the case y = 0 the semicircle distribution after appropriate rescaling. The entries we consider are Curie-Weiss spins, which are correlated random signs, where the degree of the correlation is governed by an inverse temperature beta > 0. The model exhibits a phase transition at beta = 1. The correlation between any two entries is of order O((np)(-1)) for beta is an element of (0, 1), O ((np)(-1/2)) for beta = 1, and for beta > 1 the correlation does not vanish in the limit. In our proofs we use Stieltjes transforms and concentration of random quadratic forms.

引用

页码：857 / 876

页数：20