Norms of sub-exponential random vectors

TL;DR

This paper explores different Luxemburg norms in spaces of sub-exponential random vectors, establishing relations between them and applying these norms to analyze chaos phenomena in such vectors.

Contribution

It introduces and compares various Luxemburg norms for sub-exponential vectors and applies these norms to study chaos in random vectors with sub-exponential coordinates.

Findings

Equivalent relations between different Luxemburg norms are established.

Conditions under which the uniform norm is bounded by coordinate norms are identified.

Application of norms to analyze chaos in sub-exponential random vectors.

Abstract

We discuss various forms of the Luxemburg norm in spaces of random vectors with coordinates belonging to the classical Orlicz spaces of exponential type. We prove equivalent relations between some kinds of these forms. We also show when the so-called uniform norm is majorized by norms of coordinates up to some constants. We give an application of other norm to study of chaos in random vectors with sub-exponential coordinates.