Related papers: Recent progress on Bernoulli convolutions

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We…

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_\lambda$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^n$, where the signs are independent unbiased coin tosses. We prove that each…

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…

For $(\lambda_{1},...,\lambda_{d})=\lambda\in(0,1)^{d}$ with $\lambda_{1}>...>\lambda_{d}$, denote by $\mu_{\lambda}$ the Bernoulli convolution associated to $\lambda$. That is, $\mu_{\lambda}$ is the distribution of the random vector…

One can consider $\mu$-Martin-L\"of randomness for a probability measure $\mu$ on $2^{\omega}$, such as the Bernoulli measure $\mu_p$ given $p \in (0, 1)$. We study Bernoulli randomness of sequences in $n^{\omega}$ with parameters $p_0,…

Let $\mu_{\lambda}$ be the Bernoulli convolution measure with parameter $\lambda\in(0,1)$. We study the regularity of the function %We prove that $h=h_{\phi}:\lambda\mapsto \int_{\mathbb{R}}\phi(x)\,d\mu_{\lambda}(x)$ for H\"older…

Let $\nu_\lambda^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known…

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1}…

We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.

In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-L\"of random with respect to a Bernoulli measure $\mu_p$ for some $p\in[0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in…

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi…

The random beta-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special…

Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss…

Let $\lambda\in (1,\sqrt{2}]$ be an algebraic integer with Mahler measure $2.$ A classical result of Garsia shows that the Bernoulli convolution $\mu_\lambda$ is absolutely continuous with respect to the Lebesgue measure with a density…

Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $\lambda := p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the…

The paper is devoted to infinite Bernoulli convolutions generated by positive multigeometric series and to probability distributions of random variables whose digits in an even integer base-$s$ expansion with two redundant digits form a…

In this work, Bernoulli's Law of Large Numbers, also known as the Golden theorem, has been extended to study the relations between empirical probability and empirical randomness of an otherwise random experiment. Using the example of a coin…

Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinatorial structure. Visualizing this structure we discuss…

We investigate variance bounds under symmetry constraints in classical, free, and Boolean probability, focusing on Bernoulli distributions and their noncommutative analogues, projections with trace \(p\). We show that symmetrizers under…

In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence…