Effect of truncation on large deviations for heavy-tailed random vectors

TL;DR

This paper investigates how truncating heavy-tailed random vectors affects their large deviations behavior, revealing two regimes based on the growth rate of the truncation threshold, which determine whether heavy tails are preserved or lost.

Contribution

It characterizes the impact of truncation on large deviations for heavy-tailed vectors, identifying two distinct regimes depending on the truncation growth rate.

Findings

Two regimes identified: heavy tails retained or lost depending on truncation growth rate.

Truncation threshold growth rate critically influences large deviation behavior.

Results applicable to high-dimensional heavy-tailed data analysis.

Abstract

This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the same is lost.