LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

The Equivalence of Some Bernoulli Convolutions to Lebesgue Measure

Since the 1930’s many authors have studied the distribution νλ of the random series Yλ = ∑±λn where the signs are chosen independently with probability (1/2, 1/2) and 0 < λ < 1. Solomyak recently proved that for almost every λ ∈ [ 1 2 , 1], the distribution νλ is absolutely continuous with respect to Lebesgue measure. In this paper we prove that νλ is even equivalent to Lebesgue measure for alm...

Multifractal Structure of Bernoulli Convolutions

Let ν λ be the distribution of the random series ∑∞ n=1 inλ , where in 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 (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of ν λ for typical λ. Namely, we investigate the size of the sets ∆λ,p(α) = { x ∈ R : lim r↘0 log ν λ(B(x, r)) log r =...

the investigation of the relationship between type a and type b personalities and quality of translation

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Spectra of Bernoulli Convolutions

Image Measures of Infinite Product Measures and Generalized Bernoulli Convolutions

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 ran...