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On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

Tomáš Mrkvička, Jan Rataj (2009)

Kybernetika

A method of estimation of intrinsic volume densities for stationary random closed sets in ℝ d based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the...

On invariants of random planar endomorphisms

Teimuraz Aliashvili (2003)

Banach Center Publications

We estimate the expected value of the gradient degree of certain Gaussian random polynomials in two variables and discuss its relations with some other numerical invariants of random polynomials

On random fractals with infinite branching: definition, measurability, dimensions

Artemi Berlinkov (2013)

Annales de l'I.H.P. Probabilités et statistiques

We investigate the definition and measurability questions of random fractals with infinite branching, and find, under certain conditions, a formula for the upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.

On set covariance and three-point test sets

Jan Rataj (2004)

Czechoslovak Mathematical Journal

The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of ℝ d is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic...

On testing of general random closed set model hypothesis

Tomáš Mrkvička (2009)

Kybernetika

A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set Ξ ⊆ ℝ d with values in the extended convex ring is introduced. The method is based on the summary statistics – normalized intrinsic volumes densities of the ε -parallel sets to Ξ . The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation...

On the decomposition of particle size distribution in the extraction replica method

Vratislav Horálek (1981)

Aplikace matematiky

This paper deals with the method for evaluating exposures of nickel alloy structures containing both extracted and sectioned particles. The presented stereological model makes it possible to estimate two unknown spatial parameters, the mean value of the particle size distribution and the depth of etching with the use of the information obtained from the combined structure of the exposures.

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in ℝ n i , i ≤ m, then for every F in the Grassmannian G N , n , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, π F ( μ ₁ ⊗ ⋯ ⊗ μ ₘ ) , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

On the singular values of random matrices

Shahar Mendelson, Grigoris Paouris (2014)

Journal of the European Mathematical Society

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on ℝ n that is supported in a relatively small ball and linear functionals are uniformly bounded in L p for some p > 8 , in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c n / N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure...

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