Hyperfinite law of large numbers

The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of various multiplicative properties of random variables. Introduction. In probability theory and its applications, it is usual to study a large number of random variables with low intercorrelation. Since the continuum is commonly used to model the ideal situations for macroscopic phenomena, it is natural to consider a continuum of random variables (simply called a process) with low intercorrelation. In applications (especially in economic applications), exact equality between the mean (or distribution) of a sample function and the theoretical mean (or distribution) is obtained by assuming that a continuum of independent and identically distributed random variables satisfies the law of large numbers (see for example, [1], [2], [6], and [10]). This is often referred to as “aggregation removes individual uncertainty”. There are, however, serious difficulties with this assumption. It has been pointed out by Doob that such a process, typically indexed by a Lebesgue interval, is not measurable and even has no measurable standard modification with respect to the relevant product measure (unless, of course, the common distribution is concentrated at a single point; see [3], p. 67). In fact, as noted in [6], there is a naturally constructed process with independent and identically distributed random variables (indexed by the unit Lebesgue interval) for which it is not true that almost all sample functions are measurable. If one considers a natural extension of the probability measure on the sample space so that the measurability problem disappears, then the set of sample realizations satisfying the property that the expectation or Received December 20, 1995, and in revised form, February 1, 1996. 1991 Mathematics Subject Classification. Primary 03H05, 60F15; Secondary 60A05, 60G07. The author is indebted to Yu Kiang Leong, Peter Loeb and Ali Khan for many helpful suggestions. The research is partially supported by NUS grant RP3920641. c © 1996, Association for Symbolic Logic 1079-8986/96/0202-0003/$2.00