Hotelling and Pabst  showed that the rank correlation coefficient had a limiting normal distribution under the equally likely permutations of the hypothesis of independence. Wald and Wolfowitz  developed a general theorem of this type, and Noether  and Hoeffding  have relaxed the conditions used therein. In this paper a vector form of the theorem is proved along the lines used in an example by Wald and Wolfowitz  but taking account of the singular cases in which the correlations approach one.
D. A. S. Fraser. "A Vector Form of the Wald-Wolfowitz-Hoeffding Theorem." Ann. Math. Statist. 27 (2) 540 - 543, June, 1956. https://doi.org/10.1214/aoms/1177728279