Related papers: Selfsimilar random fractal measure using contracti…
Guided by the theory of graph limits, we investigate a variant of the cut metric for limit objects of sequences of discrete probability distributions. Apart from establishing basic results, we introduce a natural operation called {\em…
We show that any equicontractive, self-similar measure arising from the IFS of contractions $(S_{j})$, with self-similar set $[0,1]$, admits an isolated point in its set of local dimensions provided the images of $S_{j}(0,1)$ (suitably)…
This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The…
We consider a complete metric space $(X,d)$ and a countable number of contractive mappings on $X$, $\mathcal{F}=\{F_i:i\in\mathbb N\}$. We show the existence of a {\em smallest} invariant set (with respect to inclusion) for $\mathcal{F}$.…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…
In this article, we construct semiparametrically efficient estimators of linear functionals of a probability measure in the presence of side information using an easy empirical likelihood approach. We use estimated constraint functions and…
A class of simplified measures is constructed to capture the key features of generic spatio-temporally chaotic systems. A combined analytical and numerical investigation allows us to extablish the scaling beahviour of the fractal dimension…
In this paper, we consider a certain convolutional Laplacian for metric measure spaces and investigate its potential for the statistical analysis of complex objects. The spectrum of that Laplacian serves as a signature of the space under…
We present dynamic measurements of a lattice gas during phase separation, which we use as an analogy for self-assembly of equilibrium ordered structures. We use two approaches to quantify the degree of 'reversibility' of this process:…
Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data and with compositional data, like percentages and the like. If the natural measure of difference is not the absolute…
Any space-filling packing of spheres can be cut by a plane to obtain a space-filling packing of disks. Here, we deal with space-filling packings generated using inversive geometry leading to exactly self-similar fractal packings. First, we…
The distribution function of a random distance in three dimensions is given and some new three-dimensional d2-tests of randomness are suggested. We show that our test statistics are not correlated with the usual test statistics and are…
The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning. Established comparison methods treat the standard setting that the distributions are supported in the same space. Recently,…
We provide a self-contained exposition of the well-known multifractal formalism for self-similar measures satisfying the strong separation condition. At the heart of our method lies a pair of quasiconvex optimization problems which encode…
One of the important consequences of the Banach Fixed Point Theorem is Hutchinson's theorem which states the existence and uniqueness of fractals in complete metric spaces. The aim of this paper is to extend this theorem for semimetric…
We address the problem of testing hypotheses about a specific value of the Fr\'echet mean in metric spaces, extending classical mean testing from Euclidean spaces to more general settings. We extend an Euclidean testing procedure…
This paper investigates new properties concerning the multifractal structure of a class of statistically self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random…
This paper investigates the estimation of the self-similarity parameter in fractional processes. We re-examine the Kolmogorov-Smirnov (KS) test as a distribution-based method for assessing self-similarity, emphasizing its robustness and…
We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions, random walks, resistor networks, and Gaussian…
We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called SSC and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show…