XX^T matrices with independent entries

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
A. Bose, Priyanka Sen
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引用次数: 2

Abstract

Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments and $p/n \to y\neq 0$, then the limiting spectral distribution (LSD) of $\frac{1}{n}S$ converges to a Mar$\check{\text{c}}$enko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of $S$ in probability or almost surely, and in particular, many of the above results follow as special cases. At the same time several new LSD results also follow from our general result. The moments of the LSD are quite involved but can be described via a set of partitions. Unlike in the i.i.d. entries case, these partitions are not necessarily non-crossing, but are related to the special symmetric partitions which are known to appear in the LSD of (generalised) Wigner matrices with independent entries. We also investigate the existence of the LSD of $S_{A}=AA^T$ when $A$ is the $p\times n$ symmetric or the asymmetric version of any of the following four random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of $\frac{1}{n}S_{A}$ for the above four cases have been studied by Bose, Gangopadhyay and Sen in 2010, when the entries are i.i.d. We show that under some general assumptions on the entries of $A$, the LSD of $S_{A}$ exists and this result generalises the existing results significantly.
有独立元素的XX^T矩阵
设$S=XX^T$为(未缩放的)样本协方差矩阵,其中$X$为具有独立条目的实数$p \times n$矩阵。众所周知,如果$X$的分量是独立同分布的,并且有足够的矩量和$p/n \to y\neq 0$,那么$\frac{1}{n}S$的极限谱分布(LSD)收敛到Mar $\check{\text{c}}$ enko-Pastur定律。这个结果的几个扩展也是已知的。我们在概率上或几乎肯定地证明了$S$的LSD的存在性的一个一般结果,特别地,上面的许多结果可以作为特例来遵循。同时,几个新的LSD结果也遵循了我们的一般结果。LSD的瞬间是相当复杂的,但可以通过一组分区来描述。与i.i.d条目的情况不同,这些分区不一定是非交叉的,而是与已知出现在具有独立条目的(广义)Wigner矩阵的LSD中的特殊对称分区有关。我们还研究了$S_{A}=AA^T$的LSD的存在性,当$A$是以下四种随机矩阵:反向循环、循环、Toeplitz和Hankel中的任意一个的$p\times n$对称或非对称版本时。Bose, Gangopadhyay和Sen在2010年研究了上述四种情况$\frac{1}{n}S_{A}$的LSD,当条目是id时。我们表明,在对$A$条目的一些一般假设下,$S_{A}$的LSD是存在的,这一结果显著推广了现有的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
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.
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