Let $ \{ X, X_{ k,i}; i \geq 1, k \geq 1 \}$ be a double array of nondegenerate i.i.d. random variables and let $\{ p_{ n}; n \geq 1 \}$ 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 $L_{ n} = \max_{ 1 \leq i < j \leq p_{ n}} \left | \hat{ \rho}^{ (n)}_{ i,j} \right |$ of the sample correlation matrix ${ \bf \Gamma}_{ n} = \left ( \hat{ \rho}_{ i,j}^{ (n)} \right )_{ 1 \leq i, j \leq p_{ n}}$ where $\hat{ \rho}^{ (n)}_{ i,j}$ denotes the Pearson correlation coefficient between $(X_{ 1, i},..., X_{ n,i})'$ and $(X_{ 1, j},..., X_{ n,j})'$. 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 \pi}} e^{ -t/2} \right \}, - \infty < t < \infty \end{ align*} where $F(x) = \mathbb{ P}(|X| \leq x), x \geq 0$ and $a_{ n} = 4 \log p_{ n} - \log \log p_{ n}$, $n \geq 2$. To establish this result, we present six interesting new lemmas which may be beneficial to the further study of the sample correlation matrix.
Categories: math.PR