Properties of complex-valued power means of random variables and their applications

TL;DR

This paper investigates complex-valued power means of i.i.d. non-integrable random variables, establishing limit theorems, integrability conditions, and demonstrating their use as robust estimators for Cauchy distribution parameters.

Contribution

It introduces the analysis of complex-valued power means for non-integrable variables and proves their limit behaviors and robustness as estimators.

Findings

Established integrability conditions for complex power means.

Proved limit theorems for the variance of the power mean.

Demonstrated robustness of complex power means as estimators for Cauchy parameters.

Abstract

We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution.