On the Asymptotic Distribution of Quadratic Forms in Uniform Order Statistics

Abstract

The asymptotic distribution of quadratic forms in uniform order statistics is studied under contiguous alternatives. Using minimal conditions on the sequence of forms, the limiting distribution is shown to be the convolution of a sum of weighted noncentral chi-squares and a normal variate. The results give approximate distribution theory even when no limit exists. As an example, high-order spacings statistics are shown to have trivial asymptotic power unless the order of the spacings grows linearly with the sample size. The results are derived from a modification of an invariance principle for quadratic forms due to Rotar, which we prove by martingale central limit methods.