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Mathematics 2019, 7(3), 284;

Means as Improper Integrals

Code B-31, Sensor Technology & Analysis Branch, Electromagnetic and Sensor Systems Department, Naval Surface Warfare Center Dahlgren, 18444 Frontage Road Suite 328, Dahlgren, VA 22448-5161, USA
Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057-1233, USA
Author to whom correspondence should be addressed.
Received: 31 January 2019 / Revised: 23 February 2019 / Accepted: 27 February 2019 / Published: 20 March 2019
PDF [286 KB, uploaded 20 March 2019]


The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function. View Full-Text
Keywords: law of large numbers; weak or Kolmogorov mean; Abel’s theorem; mollifiers; summation methods; stable distributions law of large numbers; weak or Kolmogorov mean; Abel’s theorem; mollifiers; summation methods; stable distributions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Gray, J.E.; Vogt, A. Means as Improper Integrals. Mathematics 2019, 7, 284.

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