Power means of random variables and characterizations of distributions via fractional calculus (Q6619735)

The authors investigate fractional moments and expectations of power means of complex-valued random variables by using fractional calculus.\N\NBoth negative and positive orders of the fractional derivatives are used. The one-dimensional distributions are characterized in terms of the fractional moments without any moment assumptions.\N\NThe authors explicitly compute the expectations of the power means for both the univariate Cauchy distribution and the Poincaré distribution on the upper half-plane. It is shown that for these distributions, the expectations are invariant with respect to the sample size and the value of the power.