For rounding arbitrary probabilities on finitely many categories to rational proportions, the multiplier method with standard rounding stands out. Sainte-Lague showed in 1910 that the method minimizes a goodness-of-fit criterion that nowadays classifies as a chi-square divergence. Assuming the given probabilities to be uniformly distributed, we derive the limiting law of the Sainte-Lague divergence, first when the rounding accuracy increases, and then when the number of categories grows large. The latter limit turns out to be a Levy-stable distribution.
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