Lower-Bound Estimates for Poisson-Type Approximations

Abstract

Let S = w 1 S 1 + w 2 S 2 + ⋯ + w N S N . Here S j is a sum of identically distributed random variables with weight w j > 0. We consider the cases where S j is a sum of independent random variables, the sum of independent lattice variables, or has the Markov binomial distribution. Apart from the general case, we investigate the case of symmetric random variables. Distribution of S is approximated by a compound Poisson distribution, by a second-order asymptotic expansion, and by a signed exponential measure. Lower bounds for the accuracy of approximations in uniform metric are established.