Generalized Negative Binomial Convolutions

Abstract

In Chapter 7 the class ℐ was extended to contain distributions on ℝ also. In this chapter we consider instead a class of probability distributions on $\mathbb{N}_0 = \{0, 2,\cdots\}$, this being the discrete analogue of ℐ. The analogue of the Gamma distribution is the Negative Binomial distribution. By a generalized Negative Binomial convolution (GNBC), we mean a limit distribution for a sequence of finite convolutions of Negative Binomial distributions. The class of GNBC’s is denoted ℐ_d. Many of the results valid for ℐ have their counterparts for ℐ_d, but there are also several complications.