The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n >0$ is the antiderivative of $b_{n-1}(x)$ subject to $\int_0^1 b_n(x) dx = 0$. We offer a related characterization: $b_1(x) = x - 1/2$ and $(-1)^{n-1} b_n(x)$ for $n >0$ is the $n$-fold circular convolution of $b_1(x)$ with itself. Equivalently, $1 - 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of a sum of $n$ independent random variables, each with the beta$(1,2)$ probability density $2(1-x)$ at $x \in (0,1)$. This result has a novel combinatorial analog, the {\em Bernoulli clock}: mark the hours of a $2 n$ hour clock by a uniform random permutation of the multiset $\{1,1, 2,2, \ldots, n,n\}$, meaning pick two different hours uniformly at random from the $2 n$ hours and mark them $1$, then pick two different hours uniformly at random from the remaining $2 n - 2$ hours and mark them $2$, and so on. Starting from hour $0 = 2n$, move clockwise to the first hour marked $1$, continue clockwise to the first hour marked $2$, and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked $n$ is encountered, at a random hour $I_n$ between $1$ and $2n$. We show that for each positive integer $n$, the event $( I_n = 1)$ has probability $(1 - 2^n b_n(0))/(2n)$, where $n! b_n(0) = B_n(0)$ is the $n$th Bernoulli number. For $ 1 \le k \le 2 n$, the difference $δ_n(k):= 1/(2n) - ¶( I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $δ_n( 2 n + 1 - k) = (-1)^n δ_n(k)$, which is a combinatorial analog of the well known symmetry of Bernoulli polynomials $b_n(1-x) = (-1)^n b_n(x)$.
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $μ_λ$ is the probability measure corresponding to the law of the random variable $ξ= \sum_{k=0}^\infty ξ_kλ^k$, where $ξ_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < λ< 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(μ_λ) \geq 0.96399$ for all …
The Bernoulli convolution with parameter $λ\in(0,1)$ is the measure on $\bf R$ that is the distribution of the random power series $\sum\pmλ^n$, where $\pm$ are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.
We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $ν_β$ to arbitrary given accuracy whenever $β$ is algebraic. In particular, if the Garsia entropy $H(β)$ is not equal to $\log(β)$ then we have a finite time algorithm to determine whether or not $\mathrm{dim}_\mathrm{H} (ν_β)=1$.
In this paper, we essentially compute the set of $x,y>0$ such that the mapping $z \longmapsto \Big{(}1-r+r e^z\Big{)}^x \Big{(}\dis\fracλ{λ-z}\Big{)}^{y}$ is a Laplace transform. If $X$ and $Y$ are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by $μ$ the distribution of $X+Y.$ The above problem is equivalent to finding the set of $x>0$ such that $μ^{{\ast}x}$ exists.
Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of the model has been a long standing challenge. The difficulty comes from the presence of commuting pieces and from the absence of a global clock. In this paper, we introduce and study the class of Bernoulli probability measures that we claim to be the simplest adequate probability measures on infinite traces. For this, we strongly rely on the theory of trace c…
A result of Chebyshev (1864) and Hoeffding1956}, on bounding an expectation of a given function with respect to a Bernoulli convolution (also called Poisson binomial law, or law of the number of successes in independent trials) with any given first moment, is here generalised to the case of any given first few moments, as indicated in the title. A nonprobabilistic, and perhaps more obvious, reformulation is: Every permutation invariant and separately affine-linear function of $n$ real variables $x_i\in[a,b]$ assumes its extremal values given the power sums $\sum_{i=1}^nx_i^1,\ldots, \sum_{i=1}…
The Bernoulli convolution $ν_λ$ with parameter $λ\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pmλ^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\dimν_λ=1$ for all transcendental $λ\in(1/2,1)$.
We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution. We relate moments of one infinite Bernoulli convolution with others having different but related parameters. We give examples relating Euler numbers to the moments of infinite Bernoulli convolutions. One of the examples provides moment interpretation of Pell numbers as well as new identities satisfied by Pell and Lucas numbers.
We consider a probabilistic approach to compute the Wiener--Young $Φ$-variation of fractal functions in the Takagi class. Here, the $Φ$-variation is understood as a generalization of the quadratic variation or, more generally, the $p^{\text{th}}$ variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions $Φ$ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function $Φ$, our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and li…
We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} ξ= \sum_{k=1}^\infty \frac{(-1)^{k+1}ξ_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and $\{ξ_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $ξ$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_ξ)=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. …
The Bernoulli convolution with parameter $λ\in(0,1)$ is the probability measure $μ_λ$ that is the law of the random variable $\sum_{n\ge0}\pmλ^n$, where the signs are independent unbiased coin tosses. We prove that each parameter $λ\in(1/2,1)$ with $\dimμ_λ<1$ can be approximated by algebraic parameters $ξ\in(1/2,1)$ within an error of order $\exp(-deg(ξ)^{A})$ for any number $A$, such that $\dimμ_ξ<1$. As a corollary, we conclude that $\dimμ_λ=1$ for each of $λ=\ln 2, e^{-1/2}, π/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conje…
The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semigroup generated by the maps $x\mapsto λ\cdot x\pm 1$, where $λ$ is an algebraic number. We give a bound in terms of the Mahler measure of $λ$. We also derive a bound on the dimension o…
We examine measure preserving mappings $f$ acting from a probability space $(Ω, F,μ) $ into a probability space $% (Ω^{*},F^{*},μ^{*}) ,$ where $μ^{*}=μ(f^{-1})$. Conditions on $f$, under which $f$ preserves the relations ''to be singular'' and ''to be absolutely continuous'' between measures defined on $(Ω, F) $ and corresponding image measures, are investigated. We apply the results to investigate the distribution of the random variable $% ξ=\sum\limits^{\infty}_{k=1} ξ_kλ^k,$ where $% λ\in (0;1),$ and $ξ_k$ are independent not necessarily identically distributed random variables taking the …
Max-semistable extremal behavior of AR(1)-processes connected with Bernoulli convolutions
Mathematical Institute, Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学数学研究所）；Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen（格罗宁根大学伯努利数学、计算机与人工智能研究所）
Infinite Bernoulli convolutions generated by multigeometric series and their properties
Dragomanov Ukrainian State University（德拉戈马诺夫乌克兰国立大学）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）；Central Ukrainian State University（中央乌克兰国立大学）
Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$, what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by t…
We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson $χ^2$-distance. The results are based on proper non-uniform estimates for densities. They deal with models of non-homogeneous, non-degenerate Bernoulli distributions.
We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier-Mastnak-Nica-Szpojankowski about the free Bernoulli anticommutator. We also derive economically an explicit formula for the free additive convolution of Bernoulli distributions. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure.
In this note, we study the high-temperature convolution introduced in Ref.\ \cite{mergny_cconv}, between two symmetric Bernoulli distributions. We give an analytical expression for both the Stieltjes transform and the density. This result provides the first non-trivial expression for the high-temperature convolution of two distributions and gives a new family of densities, interpolating between the centered binomial distribution with number of trials $n{=}2$ and probability of success $p{=}1/2$, and the centered and re-scaled arcsine law.
For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erd{ö}s [2] and Salem [12]. We prove weak forms of their results for general self-similar measures associated to affine contractions of the real line.
