Means as Improper Integrals

John E. Gray and Andrew Vogt
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John E. Gray: 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
Andrew Vogt: Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057-1233, USA

Mathematics, 2019, vol. 7, issue 3, 1-20

Abstract: 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.

Keywords: law of large numbers; weak or Kolmogorov mean; Abel’s theorem; mollifiers; summation methods; stable distributions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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