THE CONVEX ANALYSIS OF RANDOM VARIABLES

Any real-valued random variable induces a probability distribution on the real line which can be described by a cumulative distribution function. When the vertical gaps that may occur in the graph of that function are filled in, one gets a maximal monotone relation which describes the random variable by its characteristic curve. Maximal monotone relations in the plane are known in convex analysis to correspond to the subdifferentials of the closed proper convex functions on the real line. Here that connection is developed in terms of what those convex functions and their conjugates say about the random variables, and how that information serves in applications to stochastic optimization