On spectra of probability measures generated by GLS-expansions

Abstract

We study properties of distributions of random variables with independent identically distributed symbols of generalized Lüroth series (GLS) expansions (the family of GLS-expansions contains Lüroth expansion and $Q_{\infty }$- and $G_{\infty }^2$-expansions). To this end, we explore fractal properties of the family of Cantor-like sets $C[GLS,V]$ consisting of real numbers whose GLS-expansions contain only symbols from some countable set $V\subset N\cup \{0\}$, and derive exact formulae for the determination of the Hausdorff--Besicovitch dimension of $C[GLS,V]$. Based on these results, we get general formulae for the Hausdorff--Besicovitch dimension of the spectra of random variables with independent identically distributed GLS-symbols for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansions.

Published at http://dx.doi.org/10.15559/16-VMSTA61 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)