Fachbereich Mathematik Schwerpunkt Mathematische Statistik Und Stochastische Prozesse Bundesstr. 55, D – 20146 Hamburg Compound Binomial Approximations

We consider the approximation of the convolution product of not necessarily identical probability distributions qjI+pjF , (j = 1, . . . , n), where, for all j, pj = 1− qj ∈ [0, 1], I is the Dirac measure at point zero, and F is a probability distribution on the real line. As an approximation, we use a compound binomial distribution, which is defined in a one-parametric way: the number of trials remains the same but the pj are replaced with their mean or, more generally, with an arbitrary success probability p. We also consider approximations by finite signed measures derived from an expansion based on Krawtchouk polynomials. Bounds for the approximation error in different metrics are presented. If F is a symmetric distribution about zero or a suitably shifted distribution, the bounds have a better order than in the case of a general F . Asymptotic sharp bounds are given in the case, when F is symmetric and concentrated on two points.