ﻻ يوجد ملخص باللغة العربية
Poisson processes in the space of $(d-1)$-dimensional totally geodesic subspaces (hyperplanes) in a $d$-dimensional hyperbolic space of constant curvature $-1$ are studied. The $k$-dimensional Hausdorff measure of their $k$-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the $k$-dimensional Hausdorff measure of the $k$-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $dgeq 2$, it is shown that in case (ii) the central limit theorem holds for $din{2,3}$ and fails if $dgeq 4$ and $k=d-1$ or if $dgeq 7$ and for general $k$. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin-Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.
We consider Gaussian approximation in a variant of the classical Johnson-Mehl birth-growth model with random growth speed. Seeds appear randomly in $mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed.
We consider the probability distributions of values in the complex plane attained by Fourier sums of the form sum_{j=1}^n a_j exp(-2pi i j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If the coefficien
We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $sqrt{n}{mathcal{T}_c(P_n,Q)-mathcal{W}_c(P,Q)}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We show that the asy
We consider a class of interacting particle systems with values in $[0,8)^{zd}$, of which the binary contact path process is an example. For $d ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a
We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer