Related papers: On limiting distributions of arithmetic functions
Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number…
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like…
We propose to solve a constrained distribution steering problem, i.e., steering a stochastic linear system from an initial distribution to some final, desired distribution subject to chance constraints. We do so by characterizing the…
We give simple proofs, under minimal hypotheses, of the Weak Law of Large Numbers and the Central Limit Theorem for independent identically distributed random variables. These proofs use only the elementary calculus, together with the most…
We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings,…
We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes. In this model, each node has bounded computation and storage capabilities that do not grow with the network size.…
In this work we study loss functions for learning and evaluating probability distributions over large discrete domains. Unlike classification or regression where a wide variety of loss functions are used, in the distribution learning and…
Karlin and Altschul in their statistical analysis for multiple high-scoring segments in molecular sequences introduced a distribution function which gives the probability there are at least r distinct and consistently ordered segment pairs…
In this article, the exponentiated discrete Lindley distribution is presented and studied. Some important distributional properties are discussed. Using the maximum likelihood method, estimation of the model parameters is investigated.…
In this talk we reexamine the possibility of evaluating parton distribution functions from lattice simulations. We show that, while in principle individual moments can be extracted from lattice data, in all cases the process of…
We discuss the asymptotic behaviour of models of lattice polygons, mainly on the square lattice. In particular, we focus on limiting area laws in the uniform perimeter ensemble where, for fixed perimeter, each polygon of a given area occurs…
We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer…
This short note concerns the possible singular behaviour of moment generating functions of finite measures at the boundary of their domain of existence. We look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and…
Spherical distributions, in particular, the von Mises-Fisher distribution, are often used for problems using or modelling directional data. Since expectation and variance-covariance matrices follow from the first and second moments of the…
We consider the elephant random walk with general step distribution. We calculate the first four moments of the limiting distribution of the position rescaled by $n^\alpha$ in the superdiffusive regime where $\alpha$ is the memory…
Asymptotic properties of certain arithmetic functions involving exponential divisors are investigated.
By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a…
I develop a function that, for any integer $n \geq 2$, takes a value of 1 if $n$ is prime, 0 if $n$ is composite. I also discuss two applications: First, the characteristic function provides a new expression for the prime counting function.…
In this note we reexamine the possibility of extracting parton distribution functions from lattice simulations. We discuss the case of quasi-parton distribution functions, the possibility of using the reduced Ioffe-time distributions and…
This paper analyzes a variation on the well-known "power of two choices" allocation algorithms. Classically, the smallest of $d$ randomly-chosen options is selected. We investigate what happens when the largest of $d$ randomly-chosen…