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Means as Improper Integrals

Author

Listed:
  • 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)

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.

Suggested Citation

  • John E. Gray & Andrew Vogt, 2019. "Means as Improper Integrals," Mathematics, MDPI, vol. 7(3), pages 1-20, March.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:3:p:284-:d:215597
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