COMPOUND NEGATIVE BINOMIAL APPROXIMATIONS FOR SUMS OF
RANDOM VARIABLES

P. Vellaisamy

N. S. Upadhye

Abstract: The negative binomial approximations arise in telecommunications, network analysis
and population genetics, while compound negative binomial approximations arise, for
example, in insurance mathematics. In this paper, we first discuss the approximation of the
sum of independent, but not identically distributed, geometric (negative binomial) random
variables by a negative binomial distribution, using Kerstan’s method and the method of
exponents. The appropriate choices of the parameters of the approximating distributions are
also suggested. The rates of convergence obtained here improve upon, under certain
conditions, some of the known results in the literature. The related Poisson convergence result
is also studied. We then extend Kerstan’s method to the case of compound negative binomial
approximations and error bounds for the total variation metric are obtained. The
approximation by a suitable finite signed measure is also studied. Some interesting
special cases are investigated in detail and a few examples are discussed as well.

2000 AMS Mathematics Subject Classification: Primary: 60E05; Secondary: 62E17,
60F05.

Keywords and phrases: Compound negative binomial approximation, compound
Poisson approximation, finite signed measure, Kerstan’s method, method of exponents, total
variation distance.