On dual characterizations of continuous distributions in terms of expected values of two functions of order statistics and record values (Q1781643)

Let \(\{X_n, n \geq 1 \}\) be a sequence of i.i.d. random variables with a common continuous distribution function \(F\). Denote by \(X_{k:n}\) the order statistics of a sample \((X_1, \ldots ,X_n)\). The authors present characterizations of \(F\) in terms of expected values of two functions of the ``lower'' order statistics \(X_{n-k+1:n}\) and the \(k\)th lower record values \(Z^{(k)}_n=X_{k:L_k(n)+k-1}\), where \( \{ L_k(n),n \geq 1 \}\) is the \(k\)th lower record time sequence defined in the following manner: \(L_k(1)=1\) and, for \(n \geq 1\), \(L_k(n+1)=\min \{j>L_k(n):X_{k:L_k(n)+k-1}>X_{k:j+k- 1}\}\).