Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {Xk,i; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {pn; n ≥ 1} be a sequence of positive integers such that n/pn is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry $${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$$ of the sample correlation matrix $${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$$ where $${\hat{\rho}^{(n)}_{i,j}}$$ denotes the Pearson correlation coefficient between (X1,i, ..., Xn,i)′ and (X1,j,...,Xn,j)′. Write $${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$$ , $${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$$ , and $${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$$ . Under the assumption that $${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$$ for some δ > 0, we show that the following six statements are equivalent:$$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$$$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$$$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$$$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$$$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$$$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$where $${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$$ , and an = 4 log pn − log log pn. The equivalences between (i), (ii), (iii), and (v) assume that only $${\mathbb{E}X_{1,1}^{2} < \infty}$$ . Weak laws of large numbers for Wn and Ln, n ≥  1, are also established and these are of the form $${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$$ and $${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$$, respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of Ln obtained by Jiang. Some open problems are also posed.