Gaussian limits for generalized spacings

Details Gaussian limits for generalized spacings Gaussian limits for generalized spacings

2008-04-25

arxiv.org/abs/0804.4123v2

Annals of Applied Probability 2009, Vol. 19, No. 1, 158-185

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP537 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP537

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.

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