相关论文: Generalizing Benford's law using power laws: appli…
For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and…
Let $k \geq 2$ be an integer. We prove that factorization of integers into $k$ parts follows the Dirichlet distribution $\text{Dir}\left(\frac{1}{k},\ldots,\frac{1}{k}\right)$ by multidimensional contour integration, thereby generalizing…
The value of a social network is generally determined by its size and the connectivity of its nodes. But since some of the nodes may be fake ones and others that are dormant, the question of validating the node counts by statistical tests…
We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and…
The so-called Benford's laws are of frequent use in order to observe anomalies and regularities in data sets, in particular, in election results and financial statements. Yet, basic financial market indices have not been much studied, if…
Transmutation is a technique for extending classical probability distributions in order to give them more flexibility. In this paper, we are interested in cubic transmutations of the Pareto distribution. We establish a general formula that…
We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series…
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density…
To know the statistical distribution of a variable is an important problem in management of resources. Distributions of the power law type are observed in many real systems. However power law distributions have an infinite variance and thus…
This text provides very easy and short proofs of some basic properties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor's series, Principle of…
An introductory to generalized parton distributions is given which emphasizes their spectral property and its uses as well as the equivalence of various GPD representations. Furthermore, the status of the theory and phenomenology of hard…
Consensus about the universality of the power law feature in complex networks is experiencing profound challenges. To shine fresh light on this controversy, we propose a generic theoretical framework in order to examine the power law…
Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. Surprisingly, when exponential growth data is plotted as a simple histogram disregarding the time dimension, a remarkable…
There are now many theoretical explanations for why Benford's law of digit bias surfaces in so many diverse fields and data sets. After briefly reviewing some of these, we discuss in depth recurrence relations. As these are discrete…
In this second part of our survey on the social and natural distributions, we investigate some models, which intend to explain the statistical regularity of the natural and social distributions. There is a large variety of models and in…
The graphicality problem -- whether or not a sequence of integers can be used to create a simple graph -- is a key question in network theory and combinatorics, with many important practical applications. In this work, we study the…
Benford's law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger…
We study a well-known problem concerning a random variable $Z$ uniformly distributed between two independent random variables. Two different extensions, conditionally directed power distribution and conditionally undirected power…
The notion of density of a finite set is introduced. We prove a general theorem of set theory which refines the Gibbs, Bose--Einstein, and Pareto distributions as well as the Zipf law.
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.