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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 44, Fasc. 1,
pages 133 - 156
DOI: 10.37190/0208-4147.00175
Published online 30.9.2024
 

Power means of random variables and characterizations of distributions via fractional calculus

Kazuki Okamura
Yoshiki Otobe

Abstract:

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

2010 AMS Mathematics Subject Classification: Primary 60E10; Secondary 62E10, 26A33.

Keywords and phrases: fractional moments, power mean, characterization of distribution, Cauchy distribution, Poincaré distribution

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