## The Annals of Mathematical Statistics

### Limit Theorems Associated with Variants of the Von Mises Statistic

M. Rosenblatt

#### Abstract

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 [5]. The paper makes use of elements of the theory of stochastic processes.

#### Article information

Source
Ann. Math. Statist., Volume 23, Number 4 (1952), 617-623.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729341

Digital Object Identifier
doi:10.1214/aoms/1177729341

Mathematical Reviews number (MathSciNet)
MR52732

Zentralblatt MATH identifier
0048.36003

JSTOR