Large Deviation Principle for Occupancy Problems with Colored Balls

A Large Deviations Principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls may be distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are $r$ balls thrown into $n$ urns with the probability of a ball entering a given urn being $1/n$ (Maxwell-Boltzman statistics). The LDP applies with the scale parameter $n$ going to infinity and the number of balls increasing proportionally. It holds under mild restrictions, the key one being that the coloring process by itself satisfies a LDP. Hence the results include the important special cases of deterministic coloring patterns and of colors chosen with fixed probabilities independently for each ball.