Demimartingales

Abstract

Let $(\Omega, \mathcal{F}, P)$ be a probability space and suppose that a set of random variables X ₁,…,X _n defined on the probability space $(\Omega, \mathcal{F}, P)$ have mean zero and are associated. Let S ₀=0 and S _j =X ₁+…+X _j , j=1,…,n. Then it follows that, for any componentwise nondecreasing function g;

$$E((S_{j+1} - S_{j})g(S_1,\ldots, S_j)) \geq 0,\quad j = 1,\ldots, n$$

provided the expectation exists.