On Fine Structure of Singularly Continuous Probability Measures and Random Variables with Independent Q-Symbols (original) (raw)

$\widetilde{Q}$-representation of real numbers and fractal probability distributions

A e Q−representation of real numbers is introduced as a generalization of the s−adic expansion. It is shown that the e Q−representation is a convenient tool for the construction and study of fractals and measures with complicated local structure. Distributions of random variables ξ with independent e Q−symbols are studied in details. Necessary and sufficient conditions for the corresponding probability measures µ ξ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the e Q−representation. The metric, topological, and fractal properties for the distribution of ξ are investigated. A number of examples are presented.

$L^q$ dimensions and projections of random measures

2015

We prove preservation of L q dimensions (for 1 < q ≤ 2) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures supported on 1-variable fractals as special cases. We prove a similar result for certain convolutions, extending a result of Nazarov, Peres and Shmerkin. Recently many related results have been obtained for Hausdorff dimension, but much less is known for L q dimensions.

How projections affect the dimension spectrum of fractal measures

1997

We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension of the image of µ is min(m, D q (µ)); in particular, the q-dimension of µ is preserved provided m D q (µ). We also present results on the preservation of information dimension D 1 and pointwise dimension. Finally, for 0 q < 1 and q > 2 we give examples for which D q is not preserved by any linear transformation into R m . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.

The exact rate of convergence of the Lq-spectra of self-similar measures for q<0

Journal of Mathematical Analysis and Applications, 2008

The L q-spectrum of a Borel measure is one of the key objects in multifractal analysis, and it is widely believed that L qspectrum associated with a fractal measure encode important information about the underlying dynamics and geometry. The study of the L q-spectrum therefore plays a fundamental role in the understanding of dynamical systems or fractal measures. For q 0 Olsen [L. Olsen, Empirical multifractal moment measures and moment scaling functions of self-similar multifractals, Math. Proc. Cambridge Philos. Soc. 133 (2002) 459-485] recently determined the exact rate of convergence of the L q-spectra of a self-similar measure satisfying the Open Set Condition (OSC). Unfortunately, nothing is known about the rate of convergence for q < 0. Indeed, the problem of analysing L q-spectra for q < 0 is generally considered significantly more difficult since the L q-spectra are extremely sensitive to small variations in the distribution of μ for q < 0. The purpose of this paper is to overcome these obstacles and to investigate the more difficult problem of determining the exact rate of convergence of the multifractal L q-spectra of a self-similar measure satisfying the OSC for q < 0.

Fractal measures and their singularities: The characterization of strange sets

Nuclear Physics B - Proceedings Supplements, 1987

We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: a, which determines the strength of their singularities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of a values and the func-

Cauchy transforms of measures viewed as some functionals of Fourier transforms

In memory of Kazimierz Urbanik ABSTRACT. The Cauchy transform of a positive measure plays an important role in complex analysis and more recently in so-called free probability. We show here that the Cauchy transform restricted to the imaginary axis can be viewed as the Fourier transform of some corresponding measures. Thus this allows the full use of that classical tool. Furthermore, we relate restricted Cauchy transforms to classical compound Poisson measures, exponential mixtures, geometric infinite divisibility and free-infinite divisibility. Finally we illustrate our approach with some examples.

Typical Rényi dimensions of measures. The cases: q=1 and q=∞

Journal of Mathematical Analysis and Applications, 2007

We study the typical behaviour (in the sense of Baire's category) of the q-Rényi dimensions D μ (q) and D μ (q) of a probability measure μ on R d for q ∈ [−∞, ∞]. Previously we found the q-Rényi dimensions D μ (q) and D μ (q) of a typical measure for q ∈ (0, ∞). In this paper we determine the q-Rényi dimensions D μ (q) and D μ (q) of a typical measure for q = 1 and for q = ∞. In particular, we prove that a typical measure μ is as irregular as possible: for q = ∞, the lower Rényi dimension D μ (q) attains the smallest possible value, and for q = 1 and q = ∞ the upper Rényi dimension D μ (q) attains the largest possible value.