POISSON AND COMPOUND POISSON APPROXIMATIONS FOR RANDOM SUMS OF RANDOM-VARIABLES

We derive upper bounds for the total variation distance, d, between th e distributions of two random sums of non-negative integer-valued rand om variables. The main results are then applied to some important rand om sums, including cluster binomial and cluster multinomial distributi ons, to obtain bounds on approximating them to suitable Poisson or com pound Poisson distributions. These bounds are generally better than th e known results on Poisson and compound Poisson approximations. We als o obtain a lower bound for d and illustrate it with an example.