  Let $\mathbf{Q}=(Q_1,\ldots,Q_n)$ be a random vector drawn from the uniform
distribution on the set of all $n!$ permutations of $\{1,2,\ldots,n\}$. Let
$\mathbf{Z}=(Z_1,\ldots,Z_n)$, where $Z_j$ is the mean zero variance one random
variable obtained by centralizing and normalizing $Q_j$, $j=1,\ldots,n$. Assume
that $\mathbf {X}_i,i=1,\ldots ,p$ are i.i.d. copies of
$\frac{1}{\sqrt{p}}\mathbf{Z}$ and $X=X_{p,n}$ is the $p\times n$ random matrix
with $\mathbf{X}_i$ as its $i$th row. Then $S_n=XX^*$ is called the $p\times n$
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 $p/n\to
c\in(0,\infty)$ as $n\to\infty$. We propose a novel evaluation scheme to
estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann.
Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant
summability. In addition, we raise a two-step comparison approach to obtain the
explicit formulae for the mean and covariance functions in the CLT. Relying on
this CLT, we then construct a distribution-free 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 heavy-tailed ones.
