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.

翻译：我们考虑独立同分布（i.i.d.）非可积随机变量的幂均值。幂均值是齐次拟算术均值的一个实例。在一定条件下，幂均值满足若干极限定理，这与独立同分布可积随机变量的算术均值情形类似。我们的特色在于允许幂均值的生成函数取复数值，从而能够考虑支撑在整个实数集上的随机变量的幂均值。我们建立了独立同分布非可积随机变量幂均值的可积性，以及幂均值方差的极限定理。此外，我们还研究了幂均值随幂参数变化的行为。复值幂均值是柯西分布位置参数和尺度参数的联合无偏、强相合且稳健的估计量。