Let X₁, …, X_n be a sequence of independent real random variables with common distribution function F and density function f. Let   be the corresponding order statistics and let denote the associated spacings. Define the empirical distribution function of the spacings. It is known that G_n,F converges. We characterize completely the distributions F which give the same G_F as well as the set of G_F's when f describes the set of all densities on. Moreover, given a limiting function G, we construct all the distributions F for which G_F = G. In addition we establish two Tauberian 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 singularity at the origin).