A characterization of the weak convergence of convolution powers
Some remarks on Fourier seminorms
On set covariance and three-point test sets
The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of 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...
Sur la norme de Fourier. III.
On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane
A method of estimation of intrinsic volume densities for stationary random closed sets in 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...
Properties of distance functions on convex surfaces and applications
If is a convex surface in a Euclidean space, then the squared intrinsic distance function is DC (d.c., delta-convex) on in the only natural extrinsic sense. An analogous result holds for the squared distance function from a closed set . Applications concerning -boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.
Almost sure asymptotic behaviour of the -neighbourhood surface area of Brownian paths
We show that whenever the -dimensional Minkowski content of a subset exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in , .
Boolean cluster models: mean cluster dilations and spherical contact distances
Boolean cluster point processes with various cluster distributions are examined by means of their spherical contact distribution function. Special attention is paid to clusters with strong independence properties (Poisson clusters) and regular clusters.
The iterated version of a translative integral formula for sets of positive reach
By taking into account the work of and [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], and [Math. Nachr. 129, 67-80 (1986; Zbl 0602.52003)], [Math. Z. 205, 531-549 (1990; Zbl 0705.52006)], an integral formula is obtained here by using the technique of rectifiable currents.This is an iterated version of the principal kinematic formula for sets of positive reach and generalized curvature measures.
Convergence of randomly oscillating point patterns to the Poisson point process
Oscillating point patterns are point processes derived from a locally finite set in a finite dimensional space by i.i.d. random oscillation of individual points. An upper and lower bound for the variation distance of the oscillating point pattern from the limit stationary Poisson process is established. As a consequence, the true order of the convergence rate in variation norm for the special case of isotropic Gaussian oscillations applied to the regular cubic net is found. To illustrate these theoretical...
sequences having the same numbers of -couples of given distances
Let be a sequence with a finite number of terms equal to 1. The distance sequence of is defined as a sequence of the numbers of -couples of given distances. The paper investigates such pairs of sequences that a is different from and .
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