Asymptotic distribution of the largest off-diagonal entry of correlation matrices

Suppose that we have $n$ observations from a $p$-dimensional population. We are interested in testing that the $p$ variates of the population are independent under the situation where $p$ goes to infinity as $n\to \infty$. A test statistic is chosen to be $L_n=\max_{1\le i< j\le p}\vert\rho_{ij}\vert$, where $\rho_{ij}$ is the sample correlation coefficient between the $i$-th coordinate and the $j$-th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of $L_n$ is an extreme distribution of type $G_1$, by using the Chen-Stein Poisson approximation method and the moderate deviations for sample correlation coefficients. As a statistically more relevant result, a limit distribution for $l_n=\max_{1\le i< j\le p}\vert r_{ij}\vert$, where $r_{ij}$ is Spearman's rank correlation coefficient between the $i$-th coordinate and the $j$-th coordinate of the population, is derived.