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 linear $Φ$-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear $Φ$-variation for each function $Φ$ from our class. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.
We study a class $\mathfrak X^H$ of signed Takagi-Landsberg functions with Hurst parameter $H\in(0,1)$. We first show that the functions in $\mathfrak X^H$ admit a linear $p^{\text{th}}$ variation along the sequence of dyadic partitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can be represented as the $p^{\text{th}}$ absolute moment of the infinite Bernoulli convolution with parameter $2^{H-1}$. The existence of a continuous $p^{\text{th}}$ variation enables the use of the functions in $\mathfrak X^H$ as test integrators for higher-order pathwise Itô calculus. Our next resu…
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 $\mathrm{dim}_\mathrm{H} (ν_β)=1$.
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 $λ\in(1/2,1)$.
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 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…
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 concept of the $p^{\text{th}}$ variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator $f$. Here, we analyze the $p^{\text{th}}$ variation of a class of fractal functions, containing both the Takagi--van der Waerden and Weierstraß functions. We use a probabilistic argument to show that these functions have linear $p^{\text{th}}$ variation for a parameter $p\ge1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown moreover that if functions are constructed from (a skew…
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）
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.
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.
Applying quantitative perturbation theory for linear operators, we prove non-asymptotic limit theorems for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions X . The main results are concentration inequalities and Berry-Esseen bounds, obtained assuming neither reversibility nor `warm start' hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform X-ergodicity hypothesis, and when X consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the…
Consider the linear space of functions on the binary hypercube and the linear operator $S_δ$ acting by averaging a function over a Hamming sphere of radius $δn$ around every point. It is shown that this operator has a dimension-independent bound on the norm $L_p \to L_2$ with $p = 1+(1-2δ)^2$. This result evidently parallels a classical estimate of Bonami and Gross for $L_p \to L_q$ norms for the operator of convolution with a Bernoulli noise. The estimate for $S_δ$ is harder to obtain since the latter is neither a part of a semigroup, nor a tensor power. The result is shown by a detailed stud…
