A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN

Abstract

The aim of this paper is to provide a viable measure–theoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the `right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the Karhunen–Loéve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abundance of processes satisfying the law. Generalizations to a hyperfinite number of continuous (or discrete) parameter stochastic processes are considered. The various necessary and sufficient conditions for the validity of the law provide a rather complete understanding about the cancelation of individual risks or uncertainty in general settings. Some explicit asymptotic interpretations are also given.