On the Rate of Convergence to the Bose–Einstein Distribution

For a system of identical Bose particles sitting at integer energy levels with the probabilities of microstates given by a multiplicative measure with $\ge2$ degrees of freedom, we estimate the probability of the sequence of occupation numbers to be close to the Bose–Einstein distribution as the total energy tends to infinity. We show that a convergence result earlier proved by A. M. Vershik [Functional Anal. Appl. 30 (2), 95–105 (1996)] is a corollary of our theorems.