DISTRIBUTIONAL PROPERTIES OF THE NEGATIVE BINOMIAL LÉVY
PROCESS
Tomasz J. Kozubowski
Krzysztof Podgórski
Abstract: The geometric distribution leads to a Lévy process parameterized by the probability of
success. The resulting negative binomial process (NBP) is a purely jump and non-decreasing
process with general negative binomial marginal distributions. We review various stochastic
mechanisms leading to this process, and study its distributional structure. These results
enable us to establish strong convergence of the NBP in the supremum norm to the
gamma process, and lead to a straightforward algorithm for simulating sample
paths. We also include a brief discussion of estimation of the NPB parameters, and
present an example from hydrology illustrating possible applications of this model.
2000 AMS Mathematics Subject Classification: Primary: 60G51; Secondary: 60G50,
60E07.
Keywords and phrases: Borehole data; cluster Poisson process; compound
Poisson process; count data: Cox process; discrete Lévy process; doubly stochastic
Poisson process; fractures; gamma-Poisson process; gamma process: geometric
distribution; immigration birth process; infinite divisibility; logarithmic distribution:
over-dispersion; Pascal distribution; point process; random time transformation;
subordination; simulation.