8 Gaussian Limits for Generalized Spacings ∗

Nearest neighbor cells in Rd are used to define coefficients of divergence (φ-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. The finite-dimensional distributions of the point measures induced by the coefficients of divergence converge to those of a generalized Gaussian field with a covariance structure determined by the point densities. 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. ∗ Extended version 1 Rm 2C-323, Bell Laboratories, Lucent Technologies, 600-700 Mountain Ave, Murray Hill, NJ 07974: [email protected] 2 Department of Mathematical Sciences, University of Bath, Bath BA1 7AY, United Kingdom: [email protected] 3 Department of Mathematics, Lehigh University, Bethlehem PA 18015, USA: [email protected] 3 This material is based upon work supported by the NSA under grant H98230-06-1-0052