Large Deviations Of Sums Mainly Due To Just One Summand

TL;DR

This paper formalizes the idea that for i.i.d. heavy-tailed variables with tail index between 0 and 2, large deviations in their sum are mainly caused by a single large summand.

Contribution

It provides a rigorous formalization of the principle that large deviations in sums of heavy-tailed variables are dominated by one large term.

Findings

Large deviations are primarily due to a single summand in heavy-tailed cases.

The formalization applies to i.i.d. variables with power-like tails of index (0,2).

The results clarify the behavior of sums under heavy-tailed distributions.

Abstract

We present a formalization of the well-known thesis that, in the case of independent identically distributed random variables $X_1,\dots,X_n$ with power-like tails of index $\alpha\in(0,2)$, large deviations of the sum $X_1+\dots+X_n$ are primarily due to just one of the summands.