Characterization of the exponential distribution function by Properties of the differenceX _k+s∶n−X _k∶n of order statistics

Abstract

We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX _1∶n≤X_2∶n...≤X_n∶n be the corresponding order statistics. Put

$$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$

and

$$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho > 0,x \geqslant 0.$$

Fors=1 it is well known that each of the conditions d₁(x)=O ∀x≥0 and δ₁ (x, p) = O ∀x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond _s(x) and δ₁ (x, p), respectively.