The asymptotic distribution of a class of statistics, which has been called serial statistics, has been obtained, for permutations of the observed sample values. Specific instances of the use of such statistics, for the test of randomness of a sequence, have been given and the large sample power functions have been considered, when the alternative is a Markov process.
"Asymptotic Distribution of Serial Statistics and Applications to Problems of Nonparametric Tests of Hypotheses." Ann. Math. Statist. 25 (2) 218 - 251, June, 1954. https://doi.org/10.1214/aoms/1177728783