Characterization of the Exponential Distribution by Properties of the Difference $X_{k+s:n} – X_{k:n}$ of Order Statistics

Let X 1 , X2 , ..., X,. be independent and identically distributed random variables subject to a continuous distribution function F, let X 1 ,,X2 ,., ...,X,, be the corresponding order statistics, and write P(Xt+, Xk, ? x) P(X,_k ^! x) (x ^! 0) (0) where n.k and s are fixed integers with Ic + s n. It is an old question if condition (0) implies that F is of exponential type. In [8] we showed among others that condition (0) can be greatly relaxed; namely, it can be replaced by asymptotic relations (either as x oe or x j 0) to derive this very result. Using a theorem on integrated Cauchy functional equations and in essential way a result of [8] we find now a more elegant and deeper theorem on this subject. The case of lattice distributions is also considered and some new problems are stated.