Related papers: Locally Sampleable Uniform Symmetric Distributions

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line.

Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal…

The proliferation of agentic systems has thrust the reasoning capabilities of AI into the forefront of contemporary machine learning. While it is known that there \emph{exist} neural networks which can reason through any Boolean task…

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon, and then investigated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. We give an exact formula…

This paper presents a method to detect and recognize symmetries in Boolean functions. The idea is to use information theoretic measures of Boolean functions to detect sub-space of possible symmetric variables. Coupled with the new…

Let $\BS_1,...,\BS_n$ be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter $p\in(0,1)$. Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $0<p\le 1/2$ and $m_*(p):=1$ if…

By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein's method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying…

Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately…

A metric probability space $(\Omega,d)$ obeys the ${\it concentration\; of\; measure\; phenomenon}$ if subsets of measure $1/2$ enlarge to subsets of measure close to 1 as a transition parameter $\epsilon$ approaches a limit. In this paper…

Unbiased random vectors i.e. distributed uniformly in n-dimensional space, are widely applied and the computational cost of generating a vector increases only linearly with n. On the other hand, generating uniformly distributed random…

We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the $G(n,p)$ random graph model, with number of nodes $n$ and…

Suppose $\XX{N}$ is a uniformly distributed $N$-dimensional binary vector and $\YY{N}$ is obtained by passing $\XX{N}$ through a binary symmetric channel with crossover probability $\alpha$. Recently, Courtade and Kumar postulates that…

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same…

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function.…

Let $F$ be the cumulative distribution function (CDF) of the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$, where $q\ge2$ is an integer and $\{X_n\}_{n\geq 1}$ is a stationary stochastic process with state space $\{0,\ldots,q-1\}$. In a…

Write ${\cal F}$ for the set of homomorphisms from $\{0,1\}^d$ to ${\bf Z}$ which send $\underline{0}$ to 0 (think of members of ${\cal F}$ as labellings of $\{0,1\}^d$ in which adjacent strings get labels differing by exactly 1), and…

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a…

Consider a monotone Boolean function $f:\{0,1\}^n\to\{0,1\}$ and the canonical monotone coupling $\{\eta_p:p\in[0,1]\}$ of an element in $\{0,1\}^n$ chosen according to product measure with intensity $p\in[0,1]$. The random point…

In this paper, we establish uniform asymptotic certainty bands for the conditional cumulative distribution function. To this aim, we give exact rate of strong uniform consistency for the local linear estimator of this function. The…

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The…