The empirical distribution function and strong laws for functions of order statistics of uniform spacings (Q1067714)

For a random sample of size N from a uniform distribution over (0,1), let \(U_{n:N}\) denote the \(n^{th}\) order statistic for \(1\leq n\leq N\). Define \(U_{0:N}=0\) and \(U_{n:N}=1+U_{n-N-1:N}\) for \(n>N\). Then uniform \(m^{th}\) order spacings are given by \(D(m,n,N)=U_{n+m:N}- U_{n:N}\) for \(0\leq n\leq N.\) This paper investigates the empirical distribution function of D's, as m and N approach infinity. It obtains a Glivenko-Cantelli theorem and almost sure bounds for this function. These results are applied to obtain a strong law of large numbers for smooth linear combinations of some functions of D's.