In the present paper, an integer-valued version $(T_N)$ of the Kendall rank correlation coefficient is considered. Under the hypothesis of independence, a local limit theorem with the Edgeworth expansion for $T_N$ is proved and an asymptotic expansion of the distribution function of $T_N$ is derived.
"Asymptotic Expansion and a Local Limit Theorem for a Function of the Kendall Rank Correlation Coefficient." Ann. Statist. 4 (3) 597 - 606, May, 1976. https://doi.org/10.1214/aos/1176343465