Abstract We consider measures of covering and separation that are expressed through maxima and minima of distances between points of an hypersphere. We investigate the behavior of these measures when applied to a sample of independent and uniformly distributed points. In particular, we derive their asymptotic distributions when the number of points diverges. These results can be useful as a benchmark against which deterministic point sets can be evaluated. Whenever possible, we supplement the rigorous derivation of these limiting distributions with some heuristic reasonings based on extreme value theory. As a by-product, we provide a proof for a conjecture on the hole radius associated to a facet of the convex hull of points distributed on the hypersphere.

We consider the estimation of the entropy of a discretely-supported time series through a plug-in estimator. We provide a correction of the bias and we study the asymptotic properties of the estimator. We show that the widely-used correction proposed by Roulston (1999) is incorrect as it does not remove the$O\left ({N^{-1}}\right)$part of the bias while ours does. We provide the asymptotic distribution and we show that it differs when the values taken by the marginal distribution of the process are equiprobable (a situation that we calldegeneracy) and when they are not. We introduce estimators of the bias, the variance and the distribution under degeneracy and we study the estimation error. Finally, we propose a goodness-of-fit test based on entropy and give two motivations for it. The theoretical results are supported by specific numerical examples.