{"title":"Joint convergence of sample cross-covariance matrices","authors":"M. Bhattacharjee, A. Bose, Apratim Dey","doi":"10.30757/alea.v20-14","DOIUrl":null,"url":null,"abstract":"Suppose $X$ and $Y$ are $p\\times n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n \\to \\infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $\\rho$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, p\\to \\infty, p/n\\to y\\neq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $\\rho$. Independent copies of such matrices with same $p$ but different $n$, say $\\{n_l\\}$, different correlations $\\{\\rho_l\\}$, and different non-zero $y$'s, say $\\{y_l\\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\\sqrt{np^{-1}}(C-\\rho I_p)$ converges to an elliptic variable with parameter $\\rho^2$. In particular, this elliptic variable is circular when $\\rho=0$ and is semi-circular when $\\rho=1$. If we take independent $C_l$, then the matrices $\\{\\sqrt{n_lp^{-1}}(C_l-\\rho_l I_p)\\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/alea.v20-14","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 2
Abstract
Suppose $X$ and $Y$ are $p\times n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n \to \infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $\rho$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, p\to \infty, p/n\to y\neq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $\rho$. Independent copies of such matrices with same $p$ but different $n$, say $\{n_l\}$, different correlations $\{\rho_l\}$, and different non-zero $y$'s, say $\{y_l\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\sqrt{np^{-1}}(C-\rho I_p)$ converges to an elliptic variable with parameter $\rho^2$. In particular, this elliptic variable is circular when $\rho=0$ and is semi-circular when $\rho=1$. If we take independent $C_l$, then the matrices $\{\sqrt{n_lp^{-1}}(C_l-\rho_l I_p)\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.