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Abstract

Consider an experiment with $d+1$ possible outcomes, $d$ of which occur with probabilities $x_1,\ldots ,x_d$. If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x_1,\ldots ,x_d$. We characterize those polynomials which arise as the probability of such an event. We use this to characterize those $\vec{x}$ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.