Limit theorems for Bajraktarevi\'c and Cauchy quotient means of independent identically distributed random variables

TL;DR

This paper establishes limit theorems such as strong laws of large numbers and central limit theorems for various Cauchy quotient means of i.i.d. random variables, revealing different asymptotic behaviors and normality conditions.

Contribution

It introduces new asymptotic results for Bajraktarević, Gini, exponential, and logarithmic Cauchy quotient means, highlighting their distinct convergence properties.

Findings

Exponential and logarithmic Cauchy quotient means are asymptotically normal.

Multiplicative Cauchy quotient means require time-dependent centering for normal limits.

Geometric means exhibit similar asymptotic normality as exponential- and logarithmic Cauchy means.

Abstract

We derive strong laws of large numbers and central limit theorems for Bajraktarevi\'c, Gini and exponential- (also called Beta-type) and logarithmic Cauchy quotient means of independent identically distributed (i.i.d.) random variables. The exponential- and logarithmic Cauchy quotient means of a sequence of i.i.d. random variables behave asymptotically normal with the usual square root scaling just like the geometric means of the given random variables. Somewhat surprisingly, the multiplicative Cauchy quotient means of i.i.d. random variables behave asymptotically in a rather different way: in order to get a non-trivial normal limit distribution a time dependent centering is needed.