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.

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Keywords

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

MSC2010

11K55 60G30

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