On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Sydoruk, Liliia; Torbin, Grygoriy

doi:10.15559/17-VMSTA87

Modern Stochastics: Theory and Applications

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Volume 4, Issue 3 (2017), pp. 273–283

Liliia Sydoruk Grygoriy Torbin

https://doi.org/10.15559/17-VMSTA87

Pub. online: 26 October 2017 Type: Research Article

Open Access

Received
13 August 2017

Revised
3 October 2017

Accepted
3 October 2017

Published
26 October 2017

Abstract

The paper is devoted to the restricted Oppenheim expansion of real numbers ($\mathit{ROE}$), which includes already known Engel, Sylvester and Lüroth expansions as partial cases. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $\mathit{ROE}$-expansion contain arbitrary digit i only finitely many times. Main results of the paper state the singularity (w.r.t. the Lebesgue measure) of the distribution of a random variable with i.i.d. increments of symbols of the restricted Oppenheim expansion. General non-i.i.d. case is also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

References

[1]

Albeverio, S., Torbin, G.: Fractal properties of singularly continuous probability distributions with independent ${q}^{\ast }$-digits. Bulletin des Sciences Mathématiques 4, 356–367 (2005) MR2134126

[2]

Albeverio, S., Torbin, G.: On fine fractal properties of generalized infinite Bernoulli convolutions. Bulletin des Sciences Mathématiques 8, 711–727 (2008) MR2474489. doi:10.1016/j.bulsci.2008.03.002

[3]

Albeverio, S., Pratsiovytyi, M., Torbin, G.: Fractal probability distributions and transformations preserving the Hausdorff-Besicovitch dimension. Ergodic Theory and Dynamical Systems 1, 1–16 (2004) MR2041258. doi:10.1017/S0143385703000397

[4]

Albeverio, S., Pratsiovytyi, M., Torbin, G.: Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits. Ukr. Math. J. 9, 1361–1370 (2005) MR2216038. doi:10.1007/s11253-006-0001-0

[5]

Albeverio, S., Pratsiovytyi, M., Torbin, G.: Topological and fractal properties of subsets of real numbers which are not normal. Bulletin des Sciences Mathématiques 8, 615–630 (2005) MR2166730. doi:10.1016/j.bulsci.2004.12.004

[6]

Albeverio, S., Koshmanenko, V., Pratsiovytyi, M., Torbin, G.: Spectral properties of image measures under the infinite conflict interactions. Positivity 1, 39–49 (2006) MR2223583. doi:10.1007/s11117-005-0012-3

[7]

Albeverio, S., Koshmanenko, V., Pratsiovytyi, M., Torbin, G.: On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols. Meth. of Func. An. Top. 2, 97–111 (2011) MR2849470

[8]

Albeverio, S., Kondratiev, Y., Nikiforov, R., Torbin, G.: On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions. Bulletin des Sciences Mathématiques 3, 440–455 (2014) MR3206478. doi:10.1016/j.bulsci.2013.10.005

[9]

Albeverio, S., Garko, I., Ibragim, M., Torbin, G.: Non-normal numbers: Full Hausdorff dimensionality vs zero dimensionality. Bulletin des Sciences Mathématiques 2, 1–19 (2017) MR3614114. doi:10.1016/j.bulsci.2016.04.001

[10]

Albeverio, S., Kondratiev, Y., Nikiforov, R., Torbin, G.: On new fractal phenomena connected with infinite linear ifs. Math. Nachr. 8-9, 1163–1176 (2017) MR3666991

[11]

Albeverio, S., Goncharenko, Y., Pratsiovytyi, M., Torbin, G.: Convolutions of distributions of random variables with independent binary digits. Random Operators and Stochastic Equations 1, 89–99 (2007) MR2316190. doi:10.1515/ROSE.2007.006

[12]

Baranovskyi, O., Pratsiovytyi, M., Torbin, G.: Ostrogradskyi-Sierpinski-Pierce Series and Their Applications. Naukova Dumka, Kyiv (2013)

[13]

Fan, A.H., Wu, J.: Metric properties and exceptional sets of the oppenheim expansions over the field of laurent series. Constructive Approximation 4, 465–495 (2004) MR2078082

[14]

Galambos, J.: The ergodic properties of the denominators in the Oppenheim expansion of real numbers into infinite series of rationals. The Quarterly Journal of Mathematics 2, 177–191 (1970) MR0258777. doi:10.1093/qmath/21.2.177

[15]

Galambos, J.: Representation of Real Number by Infinite Series. Lecture notes in Math., New York/Berlin (1976)

[16]

Garko, I., Torbin, G.: On $x-q_{\infty }$-expansion of real numbers and related problems. Proceedings of International Conference “Asymptotic methods in the theory of differential equations”, 48–50 (2012)

[17]

Garko, I., Nikiforov, R., Torbin, G.: G-isomorphism of systems of numerations and faithfulness of systems of coverings. II. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 2, 6–18 (2014)

[18]

Garko, I., Nikiforov, R., Torbin, G.: On g-isomorphism of systems of numerations and faithfulness of systems of coverings. I. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 1, 120–134 (2014)

[19]

Garko, I., Nikiforov, R., Torbin, G.: On g-isomorphism of probabilistic and dimensional theories of real numbers and fractal faithfulness of systems of coverings. Probability theory and Mathematical Statistics 94, 16–35 (2016) MR3553451

[20]

Hetman, B.: Metric-topological and Fractal Theory of Representation of Real Numbers by Engel Series. Extended Abstract of PhD Thesis: 01.01.06. Institute for mathematics, Kyiv (2012)

[21]

Ivanenko, G., Lebid, M., Torbin, G.: On the Lebesgue structure and fine fractal properties of some class of infinite Bernoulli convolutions with essential overlaps. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 2, 47–60 (2012)

[22]

Ivanenko, G., Nikiforov, R., Torbin, G.: Ergodic approach to investigations of singular probability measures. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 7, 126–142 (2006)

[23]

Jessen, B., Wintner, A.: Distribution function and Riemann zeta-function. Trans. Amer. Math. Soc. 38, 48–88 (1938) MR1501802. doi:10.2307/1989728

[24]

Jun, W.: The oppenheim series expansions and Hausdorff dimensions. Acta Arithmetica 107(4), 345–355 (2003)

[25]

Levy, P.: Sur les series dont les termes sont des variables independantes. Studia Math. 3, 119–155 (1931)

[26]

Lupain, M.: The superpositions of absolutely continuous and singular continuous distribution functions. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 2, 128–138 (2012)

[27]

Lupain, M.: Fractal properties of spectra of random variables with independent identically distributed gls-symbols. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 1, 279–295 (2014)

[28]

Lupain, M.: On spectra of probability measures generated by gls-expansions. Modern Stochastics: Theory and Applications 3, 213–221 (2016)

[29]

Lupain, M., Torbin, G.: On new fractal phenomena related to distributions of random variables with independent gls-symbols. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 2, 25–39 (2014)

[30]

Nikiforov, R., Torbin, G.: On the Hausdorff-Besicovitch dimension of generalized self-similar sets generated by infinite ifs. Transactions of the National Pedagogical University (Phys.-Math. Sci.) 1, 151–162 (2012)

[31]

Nikiforov, R., Torbin, G.: Fractal properties of random variables with independent $q_{\infty }$-digits. Theory Probab. Math. Stat. 86, 169–182 (2013)

[32]

Peres, Y., Schlag, W., Solomyak, B.: Sixty years of Bernoulli convolutions. Progress in Probab. 46, 39–65 (2000) MR1785620

[33]

Pratsiovyta, I., Zadnipryanyi, M.: Sylvester expansions for real numbers and their applications. Transactions of National Pedagogical Dragomanov University: 1. Phys.-Math. Sciences. 10, 174–189 (2009)

[34]

Sinelnyk, L.: On infinite Bernoulli convolutions generated by binary-lacunary sequences. Transactions of National Pedagogical Dragomanov University: 1. Phys.-Math. Sciences. 2, 55–60 (2012)

[35]

Sinelnyk, L., Torbin, G.: Asymptotic properties of Fourier-Stiltjes transform of some classes of infinite Bernoulli convolutions. Transactions of National Pedagogical Dragomanov University: 1. Phys.-Math. Sciences. 2, 229–242 (2012)

[36]

Sinelnyk, L., Torbin, G.: On family of singular probability distributions generated by some subclass of binary-lacunary sequences. Transactions of National Pedagogical Dragomanov University: 1. Phys.-Math. Sciences. 16(1), 144–152 (2014)

[37]

Solomyak, B.: On the random series $\sum \pm {\lambda }^{n}$ (an Erdös problem). Ann. of Math. 3, 611–625 (1995) MR1356783. doi:10.2307/2118556

[38]

Torbin, G.: Fractal properties of the distributions of random variables with independent q-symbols. Transactions of the National Pedagogical University (Phys.-Math. Sci.), 3, 241–252 (2002)

[39]

Torbin, G.: Multifractal analysis of singularly continuous probability measures. Ukrainian Math. J. 5, 837–857 (2005) MR2209816. doi:10.1007/s11253-005-0233-4

[40]

Torbin, G.: Properties of distributions of random variables and dynamical systems related to Ostrogradsky series of the first kind. Transactions of the National Pedagogical University (Phys.-Math. Sci.), 7, 117–125 (2006)

[41]

Torbin, G.: Probability distributions with independent q-symbols and transformations preserving the Hausdorff dimension. Theory of Stochastic Processes 13, 281–293 (2007)

[42]

Torbin, G., Pratsiovyta, I.: The singularity of the second Ostrogradskyi series. Probab.Th. Math.Stat. 81, 187–195 (2010)

[43]

Torbin, G., Pratsiovytyi, M.: Random variables with independent ${q}^{\ast }$-symbols. Institute for Mathematics of NASU: Random evolutions: theoretical and applied problems, 95–104 (1992)

[44]

Wang, B., Wu, J.: A problem of Galambos on Oppenheim series expansions. Math. Debrecen 70, 45–58 (2007)

[45]

Wang, B., Wu, J.: Approximation ratio of the digits in Oppenheim series expansion. Math. Debrecen 73, 317–331 (2008)

[46]

Zhyhareva, Y.: Singular Probability Distributions Related to Positive Lueroth Series Expansion for Real Numbers. Extended Abstract of PhD Thesis: 01.01.05. Institute for applied mathematics and mechanics, Donetsk (2014)

Full article Cited by

Open access article under the CC BY license.

Keywords

Restricted Oppenheim expansion singular probability distributions metric theory of ROE Sylvester expansion

MSC2010

11K55 60G30

Metrics

since March 2018

635

Article info
views

377

Full article
views

361

PDF
downloads

151

XML
downloads

RSS

Authors

Abstract

References

Export citation

Copy and paste formatted citation

Download citation in file