Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Means as Improper Integrals

Version 1 : Received: 31 January 2019 / Approved: 31 January 2019 / Online: 31 January 2019 (10:58:52 CET)

A peer-reviewed article of this Preprint also exists.

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


The aim of this work is to study generalizations of the notion of 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.


Law of Large Numbers; weak or Kolmogorov mean; Abel's Theorem; mollifiers; summation methods; stable distributions


Computer Science and Mathematics, Probability and Statistics

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