PROPHET INEQUALITIES FOR PRODUCTS OF NONNEGATIVE RANDOM-VARIABLES

Stopping time and supremum comparisons known as ''prophet inequalities'' are made for products of finite sequences of non-negative integrable random variables under various restrictions on the class of distributions governing these random variables. For example, it is shown that for X0 = constant > 0, and X1, X2, ..., X(n) non-negative integrable random variables, that the expected maximum of the product sequence, Y(k) = X0X1X2 ... X(k), is no more than n+1 times the value of the product sequence when stopped by non-anticipating stopping times.