  We establish some limit theorems for quasi-arithmetic means of random
variables. This class of means contains the arithmetic, geometric and harmonic
means. Our feature is that the generators of quasi-arithmetic means are allowed
to be complex-valued, which makes considerations for quasi-arithmetic means of
random variables which could take negative values possible. Our motivation for
the limit theorems is finding simple estimators of the parameters of the Cauchy
distribution. By applying the limit theorems, we obtain some closed-form
unbiased strongly-consistent estimators for the joint of the location and scale
parameters of the Cauchy distribution, which are easy to compute and analyze.
