A New Multivariate Stochastic Order, Main Properties

Abstract

Different multivariate extensions of the bidirectional order can be considered. In this paper we propose a multivariate stochastic order which under mild conditions is an extension of the bidirectional order. The new relation is a proper stochastic order in the sense that it satisfies reflexivity, transitivity and antisymmetric properties. Moreover, the new order is integral. A maximal generator of it is obtained, which is used to obtain important properties of the order as a Strassen type theorem. We also obtain a characterization of the order by construction of random vectors on the same probability space. Different properties of the order are studied as well as connections with other stochastic orders and conditions, which in conjunction with the new order lead to the stochastic equality.