Abstract: | Let X(ω) be a random element taking values in a linear space endowed with the partial order ≤; let 0 be the class of nonnegative order-preserving functions on such that, for each g∈0, Eg(X)] is defined; and let 1?0 be the subclass of concave functions. A version of Markov's inequality for such spaces in P(X ≥ x) ≤ inf0Eg(X)]/g(x). Moreover, if E(X) = ξ is defined and if Jensen's inequality applies, we have a further inequality P(X≥x) ≤ inf1Eg(X)]/g(x) ≤ inf1g(ξ)/g(x). Applications are given using a variety or orderings of interest in statistics and applied probability. |