ON THE DISTRIBUTION FUNCTION FOR THE SPACINGS

Let X(1),..., X(n) be a sequence of independent real random variables with common distribution function F and density function f. Let X(1,n) less than or equal to ... less than or equal to X(n,n) be the corresp onding order statistics and let S-i,S-n:= X(i+1,n) - X(i,n) denote the associated spacings. Define G(n,F)(x):= n(-1)# {i:nS(i,n) less than o r equal to x} the empirical distribution function of the spacings. It is known that G(n,F) converges to G(F)(x) = 1 - integral e(-xf(y)) f(y )dy. We characterize completely the distributions F which give the sam e G(F) as well as the set of G(F)'s when f describes the set of all de nsities on R. Moreover, given a limiting function G, we construct all the distributions F for which G(F) = G. In addition we establish two T auberian theorems which relate the behaviour of G(F) at infinity (resp . in 0) to the behaviour of f at infinity (resp. in 0 when f has a sin gularity at the origin).