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Given a random sample of size n from a p dimensional random vector, we are interested in testing whether the p components of the random vector are mutually independent. This is the so-called complete independence test. In the multivariate normal case, it is equivalent to testing whether the correlation matrix is an identity matrix. In this paper, we propose a one-sided empirical likelihood method for the complete independence test based on squared sample correlation coefficients. The limiting distribution for our one-sided empirical likelihood test statistic is proved to be Z^2I(Z > 0) when both n and p tend to infinity, where Z is a standard normal random variable. In order to improve the power of the empirical likelihood test statistic, we also introduce a rescaled empirical likelihood test statistic. We carry out an extensive simulation study to compare the performance of the rescaled empirical likelihood method and two other statistics. 

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi (J Theor Probab 30(1):326–364, 2017) when the rectangular matrices are square.