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Properties of complex-valued power means of random variables and their applications

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Abstract

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.

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Acknowledgement

We wish to express our gratitude to an anonymous referee for his or her comments to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Shinshu University, Matsumoto, Japan

    Y. Akaoka & Y. Otobe

  2. Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka, Japan

    K. Okamura

  3. Gunma Bank, Ltd., Maebashi, Japan

    Y. Akaoka

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  1. Y. Akaoka
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Correspondence to K. Okamura.

Additional information

A part of this paper consists of Y. A.’s master’s thesis [3].

The second author was supported by JSPS KAKENHI 19K14549 and 22K13928.

The third author was supported by JSPS KAKENHI 16K05196 and 23K03213.

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Akaoka, Y., Okamura, K. & Otobe, Y. Properties of complex-valued power means of random variables and their applications. Acta Math. Hungar. 171, 124–175 (2023). https://doi.org/10.1007/s10474-023-01372-0

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