Markov inequalities on partially ordered spaces

AbstractLet X(ω) be a random element taking values in a linear space X endowed with the partial order ≤; let G0 be the class of nonnegative order-preserving functions on X such that, for each g∈G0, E[g(X)] is defined; and let G1ņG0 be the subclass of concave functions. A version of Markov's inequality for such spaces in P(X ≥ x) ≤ infG0E[g(X)]/g(x). Moreover, if E(X) = ξ is defined and if Jensen's inequality applies, we have a further inequality P(X≥x) ≤ infG1E[g(X)]/g(x) ≤ infG1g(ξ)/g(x). Applications are given using a variety or orderings of interest in statistics and applied probability