SOME PECULIAR BOUNDARY PHENOMENA FOR EXTREMES OF RTH NEAREST NEIGHBOR LINKS

Let Dn,r denote the largest rth nearest neighbor link for n points drawn independently and uniformly from the unit d-cube Cd. We show that according as r < d or r>d, the limiting behavior of Dn,r, as n → ∞, is determined by the two-dimensional 'faces' respectively one-dimensional 'edges' of the boundary of Cd. If d = r, a 'balance' between faces and edges occurs. In case of a d-dimensional sphere (instead of a cube) the boundary dominates the asymptotic behavior of Dn,r if d ≥ 3 or if d = 2, r ≥ 3. © 1990.