Stochastic convexity on general space

This paper presents a general theory of stochastic convexity. The notions o f stochastic convexity formulated by Shaked and Shanthikumar (1988a, 1988b, 1990) are defined for general partially ordered spaces. All of the closure properties of the one-dimensional real theory are proved to be true in thi s general framework as well and results concerning the temporal convexity o f Markov chains are sharpened. Many proofs are based on new ideas, some of which also provide insightful alternatives for proofs in Shaked and Shanthi kumar (1990). The general theory encompasses the (largely one-dimensional) stochastic con vexity theory as known from these papers, and at the same time permits trea tment of multivariate multiparameter families as well as more general rando m objects. Among others, it applies to real vector spaces and yields a theo ry of stochastic convexity for random vectors and stochastic processes. We illustrate this new scope with examples and applications from queueing t heory, coverage processes, reliability and branching processes. We show tha t the virtual waiting time process of an NHPP driven./G/1 queue is stochast ically convex in the arrival intensity function, which explains the known a dverse effect of fluctuating arrival rates; that the expected size of an i. i.d, union of random sets grows concavely; that the expected utility of rep airable items under imperfect repair policies is increasing and convex in t he probabilities of successful repair.