Gaussian Limits for Generalized Spacings

Nearest neighbor cells in $R^d$ 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. 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.