A counterexample for equivalence result between tails behavior and Grand Lebesgue Spaces norms

TL;DR

This paper presents counterexamples demonstrating that tails behavior and Grand Lebesgue Spaces norms can differ for certain classes of random variables, challenging assumptions of their equivalence.

Contribution

It provides the first known counterexamples highlighting the distinction between tails behavior and Grand Lebesgue Spaces norms.

Findings

Counterexamples show non-equivalence between tails and norms

Tails behavior and norms can diverge for some random variables

Challenges existing assumptions in probability theory

Abstract

We bring in this short report (counter -) examples on order to show a difference between tails behavior and Grand Lebesgue Spaces norms for some classes of random variables.