Reverse submartingale property arising from exchangeable random variables

Abstract.

In this work, for an exchangeable sequence of random variables {X _i, i≥1}, and two nondecreasing sequences of positive integers {h _n, n≥2} and {k _n, n≥2}, where h _n+k _n≤n, ∀n≥2, we prove that {R _n,hn,kn/n, n≥2} forms a reverse submartingale sequence, where , and X _1,n≤X _2,n≤…≤X _n,n are the order statistics based on {X ₁,…,X _n}.