Strong bounds for multidimensional spacings

Abstract

Let U1, U2,..., Un be independent random vectors uniformly distributed on (0, 1)d. We define the k-th maximal spacing Δn(k) associated to U1, ...,U1 as the k-th maximal possible value of the length of a side of a d-dimensional square block in (0,1) d, which does not intersect the sample, and which cannot be enlarged without doing so. Our main result is that Δn(k)={n-1(Log n+0(Log2n))} 1/d almost surely as n→∞. Other bounds are proposed for the limiting almost sure behavior of Δn(k) as n→∞. © 1983 Springer-Verlag.