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A set of many identical interacting agents obeying a global additive constraint is considered. Under the hypothesis of equiprobability in the high-dimensional volume delimited in phase space by the constraint, the statistical behavior of a generic agent over the ensemble is worked out. The asymptotic distribution of that statistical behavior is derived from geometrical arguments. This distribution is related with the Gamma distributions found in several multi-agent economy models. The parallelism with all these systems is established. Also, as a collateral result, a formula for the volume of high-dimensional symmetrical bodies is proposed. [ABSTRACT FROM AUTHOR]

It has been shown that the approaches based on the postulate of equiprobability of all acceptable microstates of a dynamical system (the Gibbs distributions and the Boltzmann method) do not satisfy criteria for a theory to be applicable (specifically, agreement with experimental data and inherent consistency), unlike the approach based on the Boltzmann equation. An approach that satisfies these criteria and that is based on the abandonment of the equiprobability postulate is offered. Concepts of spontaneous and stimulated microstates, which imply the failure of the equiprobability of microstates, are introduced. The arguments for the following assertions are suggested: probability and, consequently, one-particle and many-particle distribution functions in principle cannot depend on time; the molecular chaos hypothesis leads to the inherent inconsistency of the Boltzmann equation; the relaxation process results in some ordering of gas particle behavior; the distinguishability of identical particles is a necessary condition to describe the behavior of ensemble particles in terms of probability. [ABSTRACT FROM AUTHOR]

Approximations were derived for the mean and variance of G[sup 2], the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution. These approximate moments, accurate to O(N[sup -3]), may be used in fitting the distribution of G[sup 2]. Extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability. The asymptotic chi-squared distribution was found to fit poorly for k >/= 4. Satisfactory results were obtained using a multiplicative correction to the chi-squared fit. More sophisticated procedures using the beta distribution produced increased accuracy, but at the cost of excessive computational labor. [ABSTRACT FROM AUTHOR]