Multivariate Compound Poisson Distributions and Infinite Divisibility

Abstract

In this note we give a multivariate extension of the proof of Ospina & Gerber (1987) of the result of Feller (1968) that a univariate distribution on the non-negative integers is infinitely divisible if and only if it can be expressed as a compound Poisson distribution.

Modeling, simulation and inference for multivariate time series of counts using trawl processes

“…However, the literature on parametric classes of multivariate infinitely divisible discrete distributions with support on N n is rather sparse. We know that any such distribution is necessarily of discrete compound Poisson type, see [12,40,42], and always has non-negatively correlated components. We will now discuss a possible parametrization based on Poisson mixtures of random additive-effect-type models.…”

Modelling, Simulation and Inference for Multivariate Time Series of Counts Using Trawl Processes

“…However, the literature on parametric classes of multivariate infinitely divisible discrete distributions with support on N n is rather sparse. We know that any such distribution necessarily is of discrete compound Poisson type, see Feller (1968), Sundt (2000), Valderrama Ospina & Gerber (1987), and always has nonnegatively correlated components. In Section 3.2.2 we will discuss a possible parametrisation based on Poisson mixtures of random additive-effect-type models.…”