Special Lecture
Independence has long been a primary focus of probability theory. There are several versions of independence in the literature, such as pairwise independence and versions of independence involving a multiple number of random variables. For a finite collection of random variables, these notions are distinct. New properties arise, however, when mass phenomena are studied. Since one cannot put a uniform probability measure on a countably infinite set, the continuum is commonly used to model a very large number of entities. The problem here, as Doob showed, is that there are measurability problems with the usual product space models when the random variables of a continuous parameter process are independent and have a common distribution. The aim of this talk is to discuss a new product space for which these measurability problems do not arise. For this space, all of the notions of independence are, in fact, almost identical to their pairwise counterpart. We show that in the usual sequential setting, these notions are asymptotically equivalent, and that various seemingly unrelated multiplicative properties of random variables can be unified. Applications include some mass phenomena in economics.