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Abstract

A probability law selects fixed-size subsets of a finite ground set, whose elements are independently colored once. We compare the histogram law conditional on that frozen coloring with the prescribed multinomial law. Under upper negative-cylinder inequalities, the normalized marginal collision sum gives a sharp criterion for expected chi-square convergence and its leading coefficient, uniformly over positive varying color probabilities and color counts. A covariance statistic gives a linear collision lower bound for expected total variation without any dependence hypothesis on the subset law. Under the upper-cylinder hypothesis, we obtain an exact convergence criterion for binary colors with varying probabilities that combines subset collisions and color rarity. For a fixed positive color vector, the same hypothesis gives total-variation convergence exactly when the collision sum vanishes. A non-real-stable example and applications to spanning trees and matroid-base rounding illustrate the scope.