Convergence for weighted sums of Luroth type random variables

TL;DR

This paper establishes asymptotic convergence results for weighted sums of Luroth-type random variables across general Oppenheim expansions, extending classical results and analyzing convergence in distribution for such sequences.

Contribution

It generalizes Vervaat's 1972 result to all Oppenheim expansions and studies convergence of weighted sums of independent variables in this context.

Findings

Asymptotic convergence for Oppenheim expansions

Extension of Vervaat's classical result

Convergence in distribution of generalized Luroth sequences

Abstract

In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Luroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Luroth random variables.