Infinitely divisible random variables have distributions that can be written as sums of countably many independent and identical distributions, i.e. a random variable $X$ is infinitely divisible if
X= ∑_(i=1)^n X_i =X_1+X_2+⋯+X_n
for any n, where the X_i are independent and identically distributed. In terms of moment generating functions – functions with special properties that are unique to each distribution – X will have the property
M_X (θ)=(M_(X_i ) (θ) )^n
These kind of random variables are interesting to explore due to their broad application to many fields and concepts.
In this thesis, I will look at an infinitely divisible bivariate gamma distribution studied in the 1969 paper “A Class of Infinitely Divisible Bivariate Gamma Distributions" by R. C. Griffiths. Throughout this paper, Griffiths obtains his conclusions by working with the moment generating function of the bivariate gamma, but does not mention how the moment generating function might arise from random variables. I will thus construct this bivariate gamma distribution through joint Poisson and gamma random variables. I will then extend this construction method to create two different infinitely divisible multivariate gamma distributions and an infinitely divisible negative binomial distribution.