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A point process approach for spatial stochastic modeling of thunderstorm cells

Kriesche B., Hess R., Schmidt V.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

471--496

In this paper we consider two different approaches for spatial stochastic modeling of thunderstorms. Thunderstorm cells are represented using germ-grain models from stochastic geometry, which are based on Cox or doubly-stochastic cluster processes. We present methods for the operational fitting of model parameters based on available point probabilities and thunderstorm records of past periods. Furthermore, we derive formulas for the computation of point and area probabilities according to the proposed germ-grain models. We also introduce a conditional simulation algorithm in order to increase the model’s ability to precisely predict thunderstorm events. A systematic comparison of area probabilities, which are estimated from the proposed models, and thunderstorm records conclude the paper.

Distributional properties of the negative binomial Lévy process

Kozubowski T. J., Podgórski K.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 1

43--71

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.

Tail probabilities for a risk process with subexponential jumps in a regenerative and diffusion environment

Palmowski Z.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

381-405

In this paper we find a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reflected Brownian motion, square functions of the Ornstein-Uhlenbeck process and splitting reflected Brownian bridges. In particular, we consider a non-Markovian intensity proces.

On approximations to generalized Cox processes

Bening V. E., Korolev V. Yu.

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 2

247--270

A refined estimate of the rate of convergence of one-dimensional distributions of nonrandomly centered generalized Cox processes to location mixtures of normal laws is presented. Asymptotic expansions for these distributions are constructed. Some estimates for the concentration functions of these distributions are proved.