Close-point spatial tests and their application to random number generators

We study statistical tests of uniformity based on the L-p-distances between the m nearest pairs of points, for n points generated uniformly over the k -dimensional unit hypercube or unit torus. The number of distinct pairs at distance no more than t, for t greater than or equal to 0, is a stochastic process whose initial part, after an appropriate transformation and as n -- > infinity, is asymptotically a Poisson process with unit rate. Convergence to this asymptotic is slow in the hypercube as soon as k exceeds 2 or 3, d ue to edge effects, but is reasonably fast in the torus. We look at the qua lity of approximation of the exact distributions of the tests statistics by their asymptotic distributions, discuss computational issues, and apply th e tests to random number generators. Linear congruential generators fail de cisively certain variants of the tests as soon as n approaches the square r oot of the period length.