Gaussian limits for generalized spacings

Yu. Baryshnikov, Mathew D. Penrose, J. E. Yukich

Published 2008-04-25, updated 2009-03-06Version 2

Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. In $d=1$, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic $k$-spacings, information gain, log-likelihood ratios and the number of pairs of sample points within a fixed distance of each other.