Kendall's rank correlation statistic $T_n = \sum i > j \operatorname{sgn} (X_i - X_j)\cdot$ sgn $(Y_i - Y_j)$ is well known to be asymptotically normally distributed under the null hypothesis of independence as the sample size $n\rightarrow\infty$. In the note it is shown that this assertion can be obtained easily from the recurrence formula $p_n(t) = (1/n) \sum^n_{k =1}p_{n - 1}(t - 2k + n + 1)$ for the probability distribution $p_n$ of $T_n$ (see Kendall (1970), e.g.). This recurrence formula implies that $T_n$ has the same distribution as a sum of $(n - 1)$ well defined independent random variables to which the Lyapunov criterion applies.
@article{1176343354,
author = {Jirina, Miloslav},
title = {On the Asymptotic Normality of Kendall's Rank Correlation Statistic},
journal = {Ann. Statist.},
volume = {4},
number = {1},
year = {1976},
pages = { 214-215},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343354}
}
Jirina, Miloslav. On the Asymptotic Normality of Kendall's Rank Correlation Statistic. Ann. Statist., Tome 4 (1976) no. 1, pp. 214-215. http://gdmltest.u-ga.fr/item/1176343354/