A FINITARY CHARACTERIZATION OF THE EWENS SAMPLING FORMULA

Abstract

The clustering of agents in the market is a typical problem dealt with in recent approaches to macroeconomic modeling, describing macroscopic variables in terms of the behavior of a large collection of microeconomic entities. Clustering is subject to many economic interpretations, often described by the Ewens Sampling Formula (ESF). In contrast with the usual complex derivations, we suggest a finitary characterization of the ESF pointing to real economic processes. Our approach is finitary in the sense that we provide a probabilistic description of a system of n individuals considered as a closed system, a population, where individuals can change attributes over time. The probability is understood as the fraction of time the system spends in the considered partition. As the ESF represents an equilibrium distribution satisfying detailed balance, some properties difficult to prove are derived in a simple way. Besides the mean distribution of the cluster sizes, we study the probabilistic time behavior of clusters, in particular the mean survival as a function of the actual size and the correlation between size and age.