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SUMMARY:Mykola Pratsiovytyi (Mykhailo Drahomanov Ukrainian State Universit
 y\; Institute of Mathematics\, Natl. Acad. Sci. Ukraine)\, Dmytro Karvatsk
 yi (Institute of Mathematics\, Natl. Acad. Sci. Ukraine)\, and Oleh Makarc
 huk (Volodymyr Vynnychenko Central Ukrainian State University)
DTSTART:20260312T133000Z
DTEND:20260312T150000Z
DTSTAMP:20260423T004132Z
UID:fran/77
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/fran/77/">In
 finite Bernoulli convolutions governed by series with Cantorval sets of su
 bsums and their fractal properties</a>\nby Mykola Pratsiovytyi (Mykhailo D
 rahomanov Ukrainian State University\; Institute of Mathematics\, Natl. Ac
 ad. Sci. Ukraine)\, Dmytro Karvatskyi (Institute of Mathematics\, Natl. Ac
 ad. Sci. Ukraine)\, and Oleh Makarchuk (Volodymyr Vynnychenko Central Ukra
 inian State University) as part of Семінар з фрактально
 го аналізу / Fractal analysis seminar\n\n\nAbstract\nIn this talk
 \, we discuss infinite Bernoulli convolutions governed by positive multige
 ometric series as well as distributions of random variables such that digi
 ts of their representation in the system with an even natural base $s$ and
  two redundant digits are independent identically distributed random varia
 bles.\n\nThe main objects of study are the random variables\n\\[\n  \\xi =
  \\sum_{n=1}^\\infty \\frac{\\xi_n}{s^n}\,\n\\]\nwhere $(\\xi_n)$ is a seq
 uence of independent random variables taking the values $0$\, $1$\, …\, 
 $s$\, $s+1$ with probabilities $p_0$\, $p_1$\, …\, $p_s$\, $p_{s+1}$\, r
 espectively ($3 < s \\in \\mathbb{N}$) and\n\\[\n  \\eta = \\sum_{n=1}^\\i
 nfty \\left[ \\frac{3\\eta_{(n-1)(m+1)+1}}{s^n} + \\sum_{j=1}^m \\frac{2\\
 eta_{(n-1)(m+1)+1+j}}{s^n} \\right]\,\n\\]\nwhere $(\\eta_n)$ is a sequenc
 e of independent identically distributed random variables taking the value
 s $0$ and $1$ with probabilities $q_0 > 0$ and $q_1 = 1 - q_0 > 0$\, respe
 ctively. We study the conditions of absolute continuity and singularity of
  the distributions of these random variables and topological\, metric\, an
 d fractal properties of their essential supports. The cases when the spect
 rum of distribution is a Cantorval are considered in detail.\n\nWhen $s = 
 4$\, we find the necessary and sufficient conditions for the distributions
  of the random variables $\\xi$ and $\\eta$ to be singular and absolutely 
 continuous\, including the case when they are supported on the Guthrie–N
 ymann Cantorval. For an arbitrary even number $s > 4$\, we find the necess
 ary and sufficient conditions for the random variable $\\xi$ to be decompo
 sed into a sum of two independent random variables such that one of them h
 as a uniform distribution on the unit interval\, which is equivalent to it
 s absolute continuity. Using the method of characteristic functions\, we o
 btain sufficient conditions of singularity and absolute continuity. For Ca
 ntorvals that are the spectra of distributions\, their structure and fract
 al properties of their boundary are studied.\n
LOCATION:https://researchseminars.org/talk/fran/77/
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