Abstract: 
Let Q = (Q1, . . . , Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2, . . . , n}. Let Z = (Z1, . . . , Zn), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj , j = 1, . . . , n. Assume that Xi , i = 1, . . . ,p are i.i.d. copies of 1/√ p Z and X = Xp,n is the p × n random matrix with Xi as its ith row. Then Sn = XX is called the p × n Spearman's rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman's rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, p = p(n) and p/n→c ∈ (0,∞) as n→∞.We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann. Statist. 36 (2008) 25532576] to bypass the socalled joint cumulant summability. In addition, we raise a twostep comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distributionfree statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavytailed ones.

Notes: 
B.Z. was supported in part by the Ministry of Education, Singapore, under Grant # ARC 14/11, and NSF of China, Grant No. 11371317.
L.L.was supported in part by a MOE Tier 2 Grant 2014T22060 and by a MOE Tier 1 Grant RG25/14
at the Nanyang Technological University, Singapore. G.P. was supported in part by Grant R155000151112 at the National University of Singapore.
