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

Suppose that we have observations from a -dimensional population. We are interested in testing that the variates of the population are independent under the situation where goes to infinity as <!-- MATH $n\to \infty$ --> . A test statistic is chosen to be <!-- MATH $L_n=\max_{1\le i< j\le p}|\rho_{ij}|$ --> <img width="200" height="41" align="MIDDLE" border="0" src="/tran/2007-359-11/S0002-9947-07-04192-X/gif-abstract0/img6.gif" alt="$ L_n=\max_{1\le i< j\le p}\vert\rho_{ij}\vert$">, where is the sample correlation coefficient between the -th coordinate and the -th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of is an extreme distribution of type , 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 <!-- MATH $l_n=\max_{1\le i< j\le p}|r_{ij}|$ --> <img width="191" height="41" align="MIDDLE" border="0" src="/tran/2007-359-11/S0002-9947-07-04192-X/gif-abstract0/img12.gif" alt="$ l_n=\max_{1\le i< j\le p}\vert r_{ij}\vert$">, where is Spearman's rank correlation coefficient between the -th coordinate and the -th coordinate of the population, is derived.