Banach spaces characterization of random vectors with exponential decreasing tails of distribution

TL;DR

This paper characterizes random vectors with exponential tail decay using Banach space representations, identifying three key types of spaces that describe their distributional properties.

Contribution

It introduces a Banach space framework for such vectors, specifically detailing exponential Orlicz, Young, and Grand Lebesgue spaces as comprehensive models.

Findings

Identifies three types of Banach spaces for exponential tail vectors

Provides a unified representation framework for these vectors

Discusses potential applications of the theoretical results

Abstract

We present in this paper the Banach space representation for the set of random finite-dimensional vectors with exponential decreasing tails of distributions. We show that there are at last three types of these multidimensional Banach spaces, i.e. which can completely describe the random vectors with exponential decreasing tails of distributions: exponential Orlicz spaces, Young spaces and Grand Lebesgue spaces. We discuss in the last section the possible applications of obtained results.