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The set of incomplete sums of the first Ostrogradsky series and anomalously fractal probability distributions on it. (English) Zbl 1199.60036

The aim of the present paper is to study topological, metrical and fractal properties of the set of incomplete sums of the first Ostrogradsky series and of the family of sets in a metric space of compact subsets with Hausdorff metric. In particular the authors managed to investigate properties of infinite Bernoulli convolutions generated by the first Ostrogradsky series. Moreover the authors managed to study the asymptotic behaviour of the absolute value of the characteristic function of random incomplete sum of the first Ostrogradsky series at infinity and to prove that this random variable has an anomalously fractal singular distribution of the Cantor type, to study fractal properties of this probability measure as well as to obtain necessary and sufficient conditions for the Hausdorff-Billingsley dimension preservation on the topological support by its distribution function. The paper is well written and a sufficient part of the results can be extended to other classes of Bernoulli convolutions generated by the first Ostrogradsky series.

MSC:

60E05 Probability distributions: general theory
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
26A30 Singular functions, Cantor functions, functions with other special properties
28A80 Fractals
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