The characterization of distributions by order statistics and record values — a unified approach

Abstract

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successivejth record valuesandassociated to an i.i.d. sequence are such thatandare independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such thatXk, nandXk+1,n–Xk, nare independent for somek≧ 2 (whereXk, nis thekth order statistic ofX1, ···,Xn) are degenerate.