A reverse submartingale property of the range (Q1327853)

Let \(X_ 1, X_ 2,\dots\) be an exchangeable sequence of random variables; \(X_{1,n} \leq X_{2,n} \leq \cdots \leq X_{n,n}\) be the order statistics based on \(\{X_ 1, \dots, X_ n\}\) and \(R_ n = X_{n,n} - X_{1,n}\) be the range. The following statements are proved: If \(E | X_ 1 | < \infty\), then \(\{n^{-1}R_ n\), \(n \geq 2\}\) is a reverse submartingale sequence. If \(E | X_ 1 |^ k < \infty\), then \(ER^ k_ j \leq (j/i)^ kER^ k_ i\), \(2 \leq i \leq j\), \(k \geq 1\).