Modern Stochastics: Theory and Applications

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This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

We extend the Poincaré–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to ${\ell }_{\mathit{p}}$-norms are given.

We extend the Poincaré–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to ${\ell }_{\mathit{p}}$-norms are given.

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.