Chaotické náhodné veličiny v aplikované pravděpodobnosti

Večeřa, Jakub

This thesis deals with modeling of particle processes. In the first part we ex- amine Gibbs facet process on a bounded window with discrete orientation distri- bution and we derive central limit theorem (CLT) for U-statistics of facet process with increasing intensity. We calculate all asymptotic joint moments for interac- tion U-statistics and use the method of moments for deriving the CLT. Moreover we present an alternative proof which makes use of the CLT for U-statistics of a Poisson facet process. In the second part we model planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. We also introduce the Takacs-Fiksel estimate and demonstrate the use of estimators in a simulation study and also using data from actin fibres from stem cells images. In the third part we study a stationary Gibbs particle process with determin- istically bounded particles on Euclidean space defined in terms of a finite range potential and an activity parameter. For small activity parameters, we prove the CLT for certain statistics of this...

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