On the limit distribution of extremes of generalized Oppenheim random variables

TL;DR

This paper studies the asymptotic behavior of the extremes of generalized Oppenheim random variables, establishing conditions for their maxima to follow the Frechet distribution and exploring their asymptotic independence.

Contribution

It provides new conditions under which the extremes of Oppenheim variables belong to the Frechet domain and proves an Extreme Types theorem for dependent structures.

Findings

Normalized extremes follow Frechet distribution under certain conditions

Maxima and minima exhibit asymptotic independence

Established an Extreme Types theorem for dependent Oppenheim expansions

Abstract

This paper investigates the asymptotic behavior of the extremes of a sequence of generalized Oppenheim random variables. Particularly, we establish conditions under which some normalized extremes of sequences arising from Oppenheim expansions belong to the maximum domain of attraction of the Frechet distribution. Additionally, we identify conditions under which the maxima and minima of Oppenheim random variables demonstrate some kind of asymptotic independence. Finally, we prove an Extreme Types theorem for Oppenheim expansions with unknown dependent structure.