Large Deviations Principle for Occupancy Problems with Colored Balls

Abstract

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There arerballs thrown intonurns with the probability of a ball entering a given urn being 1/n(i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter,n, tending to infinity andrincreasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.