On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

Abstract

Let $\{X,X_{k,i};i\ge 1,k\ge 1\}$ be a double array of nondegenerate i.i.d. random variables and let $\{p_n;n\ge 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from $0$ and $\infty$. This paper is devoted to the solution to an open problem posed in Li, Liu, and Rosalsky (2010) on the asymptotic distribution of the largest entry $L_n={\max }_{1\le i<j\le p_n}\left|{\hat{\rho }}_{i,j}^{(n)}\right|$ of the sample correlation matrix ${\mathbf{\Gamma }}_n={\left({\hat{\rho }}_{i,j}^{(n)}\right)}_{1\le i,j\le p_n}$ where ${\hat{\rho }}_{i,j}^{(n)}$ denotes the Pearson correlation coefficient between $(X_{1,i},...,X_{n,i}{)}^{\prime }$ and $(X_{1,j},...,X_{n,j}{)}^{\prime }$. We show under the assumption $\mathbb{E}{X}^2<\infty$ that the following three statements are equivalent: \begin{align*} & {\bf (1)} \quad \lim_{n \to \infty} n^{2} \int_{(n \log n)^{1/4}}^{\infty} \left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x} \right) \right) dF(x) = 0, \\ & {\bf (2)} \quad \left ( \frac{n}{\log n} \right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2, \\ & {\bf (3)} \quad \lim_{n \rightarrow \infty} \mathbb{P} \left (n L_{n}^{2} - a_{n} \leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 π}} e^{-t/2} \right \}, - \infty < t < \infty \end{align*} where $F(x)=\mathbb{P}(|X|\le x),x\ge 0$ and $a_n=4\log p_n-\log \log p_n$, $n\ge 2$. To establish this result, we present six interesting new lemmas which may be beneficial to the further study of the sample correlation matrix.

16 pages

References in corpus (2)

• The asymptotic distribution and Berry--Esseen bound of a new test for independence in high dimension with an application to stochastic optimization
• Some strong limit theorems for the largest entries of sample correlation matrices

Cited by in corpus (1)

• Phase Transition in Limiting Distributions of Coherence of High-Dimensional Random Matrices