Stein's method and multinomial approximation (Q1201313)

Stein's method [see \textit{C. Stein}, Proc. 6th Berkeley Sympos. Math. Statist. Probab., Univ. Calif. 1970, 2, 583-602 (1972; Zbl 0278.60026)] is developed in the setting of approximation by a multinomial distribution \(MN(N;p_ 1,\dots,p_ M)\) for an arbitrary choice of \(M\) and \(p_ 1,\dots,p_ M\). A Stein equation is constructed, and smoothness estimates for its solutions are derived. The method is then applied in approximating three distributions: the joint distribution of the digits in the base \(M\) expansion of a random integer, the multivariate Poisson-binomial distribution and the multivariate hypergeometric distribution.