A THEORY OF HYPERFINITE PROCESSES - THE COMPLETE REMOVAL OF INDIVIDUAL UNCERTAINTY VIA EXACT LLN

The aim of this paper is to provide a viable measure-theoretic framewo rk for the study of random phenomena involving a large number of econo mic entities. The work is based on the fact that processes which are m easurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus c onstitute the 'right' class for general stochastic modeling, The prima ry 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 indepe ndence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed vi a a biorthogonal expansion of the Karhunen-Loeve type and via the repr esentation 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 stochas tic processes are considered. The various necessary and sufficient con ditions for the validity of the law provide a rather complete understa nding about the cancelation of individual risks or uncertainty in gene ral settings. Some explicit asymptotic interpretations are also given. (C) 1998 Elsevier Science S.A.