THE LIMIT DISTRIBUTION OF THE MAXIMUM PROBABILITY NEAREST-NEIGHBOUR BALL

Let X-1,...,X-n be independent random points drawn from an absolutely continuous probability measure with density f in R-d. Under mild conditions on f, we derive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by P-n the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nP(n) - ln n as n -> infinity. Moreover, we derive a tight upper bound on the upper tail of the distribution of nP(n) - ln n, which does not depend on f.