相关论文: On Sequences with Exponentially Distributed Gaps

De, Trevisan and Tulsiani [CRYPTO 2010] show that every distribution over $n$-bit strings which has constant statistical distance to uniform (e.g., the output of a pseudorandom generator mapping $n-1$ to $n$ bit strings), can be…

One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of…

The well known Three Gap Theorem states that there are at most three gap sizes in the sequence of fractional parts $\{\alpha n\}_{n<N}$ . It is known that if one averages over {\alpha}, the distribution becomes continuous. We present an…

This paper addresses the question of the fluctuations of the empirical entropy of a chain of infinite order. We assume that the chain takes values on a finite alphabet and loses memory exponentially fast. We consider two possible…

For a sequence of nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to…

We prove {\em sign equidistribution} of Legendre polynomials: the ratio between the lengths of the regions in the interval $[-1, 1]$ where the Legendre polynomial assumes positive versus negative values, converges to one as the degree…

We prove an entropy version of van der Corput's difference theorem: the entropy of a sequence is equal to the entropy of its differences. This reveals a potential correspondence between the theory of uniform distribution mod 1 and entropy.…

We consider the general branching random walk under minimal assumptions, which in particular guarantee that the empirical particle distribution admits an almost sure central limit theorem. For such a process, we study the large time decay…

The three gap theorem was originally a conjecture by Steinhaus, who asserted that there are at most three distinct gap lengths in the fractional parts of the sequence {\alpha},{2}{\alpha},{\cdots},{N}{\alpha} for any integer {N} and real…

We show that in a sample of size $n$ from a GEM$(0,\theta)$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention $G_{n:n} :=…

In this paper, we give some conditions, under which, if an infinitely divisible distribution supported on $[0,\infty)$ belongs to the intersection of exponential distribution class $\mathcal{L}(\gamma)$ for some $\gamma\ge0$ and generalised…

We establish a non-Archimedean analogue of Koksma's theorem. For a local field F of characteristic zero, we prove that the sequence ([{\alpha}x^n]) is uniformly distributed in the valuation ring O for almost every x with |x|_p>1. In the…

This paper investigates what can be inferred about an arbitrary continuous probability distribution from a finite sample of $N$ observations drawn from it. The central finding is that the $N$ sorted sample points partition the real line…

We show that an arbitrary probability distribution can be represented in exponential form. In physical contexts, this implies that the equilibrium distribution of any classical or quantum dynamical system is expressible in grand canonical…

We prove the following exponential inequality: Let $n\geq 1$ and let $X_1,...,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have \[\mathbb{P}\big({X_1+...+X_n}\geq x,\,…

A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of…

We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions $\Delta^2x_n\to…

This is an erratum to 'On the quantitative distribution of polynomial nilsequences' [GT]. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We…

The purpose of this article is to formulate a number of probabilistic hidden-variable theorems, to provide proofs in some cases, and counterexamples to some conjectured relationships. The first theorem is the fundamental one. It asserts the…

There are given characterizations of the exponential distribution by the properties of the independence of linear forms with random coefficients. Related results based on the constancy of regression of one statistic on a linear form are…