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 a homogeneous quasi-arithmetic mean, and under some conditions, several limit theorems hold for the power mean as well as for the arithmetic mean of i.i.d. integrable random variables. We establish integrabilities and a limit theorem for the variances of the power mean of i.i.d. non-integrable random variables. We also consider behaviors of the power mean when the parameter of the power varies. Our feature is that the generator of the power mean is allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. The complex-valued power mean is an unbiased strongly-consistent estimator for the joint of the location and scale parameters of the Cauchy distribution.