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From almost sure local regularity to almost sure Hausdorff dimension for gaussian fields

Erick Herbin, Benjamin Arras, Geoffroy Barruel (2014)

ESAIM: Probability and Statistics

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph...

Functional central limit theorems for seeds in a linear birth and growth model

A. Dziwisz, W. Szczotka (2016)

Applicationes Mathematicae

A problem of heredity of mixing properties (α-mixing, β-mixing and ρ-mixing) from a stationary point process on ℝ × ℝ₊ to a sequence of some of its points called 'seeds' is considered. Next, using the mixing properties, several versions of functional central limit theorems for the distances between seeds and the process of the number of seeds are obtained.

Functionals of spatial point processes having a density with respect to the Poisson process

Viktor Beneš, Markéta Zikmundová (2014)

Kybernetika

U -statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener-Itô chaos expansion. In the second half we obtain more explicit results for a system of U -statistics of some parametric models in stochastic geometry. In the logarithmic form functionals are connected to Gibbs models. There is an inequality between moments of Poisson...

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