Some distributions associated with bose-einstein statistics.

ABSTRACT

This paper examines a stochastic process for Bose-Einstein statistics that is based on Gibrat's Law (roughly the probability of a new occurrence of an event is proportional to the number of times it has occurred previously). From the necessary conditions for the steady state of the process are derived, under two slightly different sets of boundary conditions, the geometric distribution and the Yule distribution, respectively. The latter derivation provides a simpler method than the one earlier proposed by Hill [J. Amer. Statist. Ass. (1974) 69, 1017-1026] for obtaining the Pareto Law (a limiting case of the Yule distribution) from Bose-Einstein statistics. The stochastic process is applied to the phenomena of city sizes and growth.