A multidimensional analogue of the von Mises statistic is considered for the case of sampling from a multidimensional uniform distribution. The limiting distribution of the statistic is shown to be that of a weighted sum of independent chi-square random variables with one degree of freedom. The weights are the eigenvalues of a positive definite symmetric function. A modified statistic of the von Mises type useful in setting up a two sample test is shown to have the same limiting distribution under the null hypothesis (both samples come from the same population with a continuous distribution function) as that of the one-dimensional von Mises statistic. We call the statistics mentioned above von Mises statistics because they are modifications of the $\omega^2$ criterion considered by von Mises . The paper makes use of elements of the theory of stochastic processes.
"Limit Theorems Associated with Variants of the Von Mises Statistic." Ann. Math. Statist. 23 (4) 617 - 623, December, 1952. https://doi.org/10.1214/aoms/1177729341