Related papers: Taking rational numbers at random
We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of…
Given a set of independent Poisson random variables with common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with…
We consider asymptotic normality of linear rank statistics under various randomization rules met in clinical trials and designed for patients' allocation into treatment and placebo arms. Exposition relies on some general limit theorem due…
A. Renyi \cite{Renyi} made a definition that gives one generalization of simple normality in the context of $Q$-Cantor series. Similarly, in this paper we give a definition which generalizes the notion of normality in the context of…
We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.
This text presents an unified approach of probability and statistics in the pursuit of understanding and computation of randomness in engineering or physical or social system with prediction with generalizability. Starting from elementary…
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…
We study the question of identity testing for structured distributions. More precisely, given samples from a {\em structured} distribution $q$ over $[n]$ and an explicit distribution $p$ over $[n]$, we wish to distinguish whether $q=p$…
We determine a set of permutation patterns $q$ so that the number of permutations with $r$ occurrences of $q$ is asymptotically $n^r$ times the number of permutations avoiding $q$, partially settling a conjecture of Conway and Guttman. We…
We provide precise asymptotic estimates for the number of several classes of labelled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky et al.…
One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges…
We observe that a probability distribution supported by $\mathbb{N}$, induces a representation of real numbers in [0, 1) with digits in $\mathbb{N}$. We first study the Hausdorff dimension of sets with prescribed digits with respect to…
We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant…
We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in…
We consider a random interval splitting process, in which the splitting rule depends on the empirical distribution of interval lengths. We show that this empirical distribution converges to a limit almost surely as the number of intervals…
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these…
The present paper analyzes the discrepancy of distribution of rational points on general semisimple algebraic group varieties. The results include mean-square, almost sure, and uniform discrepancy estimates with explicit error bounds, which…
Building on the notion of $q$-integral introduced by Thomae in 1869, we introduce $q$-order statistics (that, is $q$-analogues of the classical order statistics, for $0<q<1$) which arise from dependent and not identically distributed…
A binary representation of complex rational numbers and their arithmetic is described that is not based on qubits. It takes account of the fact that $0s$ in a qubit string do not contribute to the value of a number. They serve only as place…
By a suitable shifting-the-mean parametrization at the Dirichlet series level and Delange's Tauberian theorems, we show that the number of factors in random ordered factorizations of integers is asymptotically normally distributed.