Stochastic dominance and weak concentration for sums of independent symmetric random vectors

TL;DR

This paper proves that superstrong domination and a form of weak concentration are preserved when summing independent symmetric random vectors, advancing understanding of their probabilistic structure.

Contribution

It demonstrates that superstrong domination and weak concentration properties are inherited under summation of independent symmetric random vectors, answering a longstanding question.

Findings

Superstrong domination is preserved under sums of independent symmetric vectors.

A certain weak concentration property is also inherited in this setting.

The results extend the understanding of probabilistic dominance and concentration in high-dimensional vectors.

Abstract

Kwapien and Woyczynski asked in their monograph (1992) whether their notion of superstrong domination is inherited when taking sums of independent symmetric random vectors (one vector dominates another if, essentially, tail probabilities of any norm of the two vectors compare up to some scaling constants). We answer this question positively. As a by-product of our methods, we establish that a certain notion of weak concentration is also preserved by taking sums of independent symmetric random vectors.