Stein's Method and Multinomial Approximation

In this paper Stein's method is considered in the context of approximation by a multinomial distribution. By using a probabilistic argument of Barbour, whereby the essential ingredients necessary for the application of Stein's method are derived, the Stein equation for the multinomial distribution is obtained. Bounds on the smoothness of its solution are derived and are used in three examples to give error bounds for the multinomial approximation to the distribution of a random vector.