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

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

Automated summary

This study investigates the power means of independent and identically distributed non-integrable random variables. It establishes several limit theorems and introduces the concept of complex-valued power means. The study also examines the behavior of power means as the power parameter varies and applies complex-valued power means to estimate the parameters of the Cauchy distribution.