We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ ℝd that is based on the maximal spacing generated by independent and identically distributed points X1, . . ., Xn in K, i.e. the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the d > 1 case are only availabe under uniformity, a key element of the proof is a suitable coupling. The proof is general enough to cover the case of testing for uniformity on compact Riemannian manifolds with spacings defined by geodesic balls.

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

Let·p denote the p-norm on . For n points drawn independently and uniformly from the unit d-cube, we obtain the limit distribution of the largest nearest-neighbour link (with respect to | · |p) in the cases ; (2) d = 3, p = 1, 2, ∞and (3) 4, p =∞.

For a sample of points drawn uniformly from either the d-dimensional torus or the d-cube, d ≧ 2, we give limiting distributions for the largest of the nearest-neighbor links. For d ≧ 3 the behavior in the torus is proved to be different from the behavior in the cube. The results given also settle a conjecture of Henze (1982) and throw light on the choice of the cube or torus in some probabilistic models of computational complexity of geometrical algorithms.

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Let X1, X2, · ··, Xn be independent identically distributed random points with common density f(x), taking values in a bounded region (p ≧ 1). We obtain the limit distribution, as n → ∞, for the maximum value of the suitably ‘weighted' (according to f(x)) rth-nearest-neighbour distances of Χ1, · ··, Χ n (r ≧ 1 fixed) provided that f(x) is bounded from below by a positive constant and a weak continuity condition holds. This is achieved by refining an argument used by the author (Henze (1981)) to derive the limit distribution in the special case r = 1. Edge-effects are eliminated by defining, for each Xi, the distance to the boundary of G to be the ‘rth-nearest-neighbour distance' if it is smaller than the distance to the rth nearest neighbour among the remaining points. Applications to a multivariate test of goodness of fit are given.

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.