## 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.

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.

Original language | English |
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Pages (from-to) | 43-71 |

Journal | Probability and Mathematical Statistics |

Volume | 29 |

Issue number | Fasc. 1 |

Publication status | Published - 2009 |

## Subject classification (UKÄ)

- Probability Theory and Statistics

## Free keywords

- 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