Averaging of random variables and fields

TL;DR

This paper proves that averaging random variables and fields does not increase their tail probabilities, under very weak conditions, ensuring stability of distribution tails during averaging.

Contribution

It introduces a nearly exact proof that averaging does not increase tail probabilities of random variables and fields under weak assumptions.

Findings

Averaging preserves tail decay rates of random variables.

The results hold under very weak conditions.

Tail probabilities do not increase significantly after averaging.

Abstract

We will prove that by averaging of random variables (r.v.) and random fields (r.f.) its tails of distributions do not increase in comparison with the tails of source variables, essentially or almost exact, under very weak conditions.