大型随机置换矩阵的极限谱分布

IF 0.7 4区 数学 Q3 STATISTICS & PROBABILITY
Jianghao Li, Huanchao Zhou, Zhidong Bai, Jiang Hu
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Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline1.png\" /> <jats:tex-math> $\\textbf X = (\\textbf x_1,\\ldots,\\textbf x_n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline2.png\" /> <jats:tex-math> $\\in \\mathbb{C} ^{m \\times n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline3.png\" /> <jats:tex-math> $m \\times n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> data matrix after self-normalization (<jats:italic>n</jats:italic> samples and <jats:italic>m</jats:italic> features), where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline4.png\" /> <jats:tex-math> $\\textbf x_j = (x_{1j}^{*},\\ldots, x_{mj}^{*} )^{*}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Specifically, we generate a permutation matrix <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline5.png\" /> <jats:tex-math> $\\textbf X_\\pi$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by permuting the entries of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline6.png\" /> <jats:tex-math> $\\textbf x_j$ </jats:tex-math> </jats:alternatives> </jats:inline-formula><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline7.png\" /> <jats:tex-math> $(j=1,\\ldots,n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and demonstrate that the empirical spectral distribution of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline8.png\" /> <jats:tex-math> $\\textbf {B}_n = ({m}/{n})\\textbf{U} _{n} \\textbf{X} _\\pi \\textbf{X} _\\pi^{*} \\textbf{U} _{n}^{*}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline9.png\" /> <jats:tex-math> $\\textbf{U} _n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a sequence of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline10.png\" /> <jats:tex-math> $p \\times m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> non-random complex matrices. The conditions we require are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline11.png\" /> <jats:tex-math> $p/n \\to c &gt;0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000081_inline12.png\" /> <jats:tex-math> $m/n \\to \\gamma &gt; 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"50 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The limiting spectral distribution of large random permutation matrices\",\"authors\":\"Jianghao Li, Huanchao Zhou, Zhidong Bai, Jiang Hu\",\"doi\":\"10.1017/jpr.2024.8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline1.png\\\" /> <jats:tex-math> $\\\\textbf X = (\\\\textbf x_1,\\\\ldots,\\\\textbf x_n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline2.png\\\" /> <jats:tex-math> $\\\\in \\\\mathbb{C} ^{m \\\\times n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline3.png\\\" /> <jats:tex-math> $m \\\\times n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> data matrix after self-normalization (<jats:italic>n</jats:italic> samples and <jats:italic>m</jats:italic> features), where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline4.png\\\" /> <jats:tex-math> $\\\\textbf x_j = (x_{1j}^{*},\\\\ldots, x_{mj}^{*} )^{*}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Specifically, we generate a permutation matrix <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline5.png\\\" /> <jats:tex-math> $\\\\textbf X_\\\\pi$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by permuting the entries of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline6.png\\\" /> <jats:tex-math> $\\\\textbf x_j$ </jats:tex-math> </jats:alternatives> </jats:inline-formula><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline7.png\\\" /> <jats:tex-math> $(j=1,\\\\ldots,n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and demonstrate that the empirical spectral distribution of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline8.png\\\" /> <jats:tex-math> $\\\\textbf {B}_n = ({m}/{n})\\\\textbf{U} _{n} \\\\textbf{X} _\\\\pi \\\\textbf{X} _\\\\pi^{*} \\\\textbf{U} _{n}^{*}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline9.png\\\" /> <jats:tex-math> $\\\\textbf{U} _n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a sequence of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline10.png\\\" /> <jats:tex-math> $p \\\\times m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> non-random complex matrices. The conditions we require are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline11.png\\\" /> <jats:tex-math> $p/n \\\\to c &gt;0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000081_inline12.png\\\" /> <jats:tex-math> $m/n \\\\to \\\\gamma &gt; 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":50256,\"journal\":{\"name\":\"Journal of Applied Probability\",\"volume\":\"50 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-04-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/jpr.2024.8\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/jpr.2024.8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

摘要

我们探讨了大维随机置换矩阵的极限谱分布,假设底层种群分布具有一般的依赖结构。让 $\textbf X = (\textbf x_1,\ldots,\textbf x_n)$$\in \mathbb{C} 是一个 $m \times n} 的数据矩阵。^{m times n}$ 是自归一化(n 个样本和 m 个特征)后的 $m times n$ 数据矩阵,其中 $\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*} )^{*}$。具体来说,我们通过对 $\textbf x_j$ (j=1,\ldots,n)$ 的条目进行置换,生成一个置换矩阵 $\textbf X_\pi$,并证明了 $\textbf {B}_n = ({m}/{n})\textbf{U} 的经验谱分布。_{n}\textbf{X} _\pi \textbf{X} _\pi^{*}\textbf{U} _{n} \textbf{X} _\pi^{*}_{n}^{*}$ 弱收敛于概率为 1 的广义马尔琴科-帕斯图分布,其中 $\textbf{U} _n$ 是$textbf{U}的序列。_n$ 是一个 $p \times m$ 非随机复矩阵序列。我们需要的条件是 $p/n \to c >0$ 和 $m/n \to \gamma > 0$ 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The limiting spectral distribution of large random permutation matrices
We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let $\textbf X = (\textbf x_1,\ldots,\textbf x_n)$ $\in \mathbb{C} ^{m \times n}$ be an $m \times n$ data matrix after self-normalization (n samples and m features), where $\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*} )^{*}$ . Specifically, we generate a permutation matrix $\textbf X_\pi$ by permuting the entries of $\textbf x_j$ $(j=1,\ldots,n)$ and demonstrate that the empirical spectral distribution of $\textbf {B}_n = ({m}/{n})\textbf{U} _{n} \textbf{X} _\pi \textbf{X} _\pi^{*} \textbf{U} _{n}^{*}$ weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where $\textbf{U} _n$ is a sequence of $p \times m$ non-random complex matrices. The conditions we require are $p/n \to c >0$ and $m/n \to \gamma > 0$ .
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来源期刊
Journal of Applied Probability
Journal of Applied Probability 数学-统计学与概率论
CiteScore
1.50
自引率
10.00%
发文量
92
审稿时长
6-12 weeks
期刊介绍: Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used. A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.
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