Means as Improper Integrals

John Gray,Andrew Vogt

doi:10.3390/math7030284

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.

Highlights

The mean is sometimes taken as the foundational principle for all of probability theory
Instead of integrating x over R with respect to the probability distribution P of the random variable X we integrate over the interval [c − M, c + M] for some choice of a center c and determine what happens as M tends to ∞
We examine a weak mean proposed by Kolmogorov that corresponds to the weak law of large numbers, and we show that it is additive

Summary

Introduction

The mean is sometimes taken as the foundational principle for all of probability theory (see, for example, [1]). Instead of integrating x over R with respect to the probability distribution P of the random variable X we integrate over the interval [c − M, c + M] for some choice of a center c and determine what happens as M tends to ∞ We consider two further generalizations: one with a weakening of the decay conditions that Kolmogorov imposed for his weak mean, which we call the doubly-weak mean; and another, the superweak mean, when the improper integral exists for at least one choice of the center c.

Improper Integrals Extending the Mean

Weak Means

Mollifiers

Cumulative Distributions and Characteristic Functions

Conclusions