[Paper Review] On exponential type Orlicz spaces of random variables

Krzysztof Zajkowski|arXiv (Cornell University)|Sep 9, 2017

Probability and Risk Models6 references1 citations

TL;DR

This paper introduces a new characterization of exponential-type Orlicz spaces generated by functions of the form $\exp\{|x|^p\} - 1$ for $p \geq 1$, leveraging this to establish a novel Bernstein-type inequality for weighted sums of independent random variables, significantly advancing tail probability bounds in non-Gaussian settings.

ABSTRACT

A new characteristics of the exponential type Orlicz spaces generated by the functions $\exp\{|x|^p\}-1$ ($p\ge 1$) is given. We use this characteristics to prove a new Bernstein-type inequality for weighted sums of independent random variables.

Motivation & Objective

• To develop a new characterization of exponential-type Orlicz spaces defined by $\exp\{|x|^p\} - 1$ functions for $p \geq 1$.
• To address the lack of sharp tail probability bounds for weighted sums of independent random variables in non-Gaussian, heavy-tailed settings.
• To derive a new Bernstein-type inequality that improves upon existing results in exponential-type Orlicz norms.

Proposed method

• The authors define and analyze Orlicz spaces using the Luxemburg norm associated with the Young function $\Psi_p(x) = \exp\{|x|^p\} - 1$ for $p \geq 1$.
• They establish equivalent norm characterizations that facilitate the analysis of moment and tail behavior of random variables in these spaces.
• Using the new norm characterization, they derive moment inequalities for weighted sums of independent random variables.
• The key technical step involves bounding the Orlicz norm of weighted sums via exponential moments, leading to a Bernstein-type inequality in the Orlicz space framework.
• The method relies on convexity and sub-Gaussian/sub-Exponential type moment estimates adapted to the $\Psi_p$-norm structure.

Experimental results

Research questions

• RQ1 How can the structure of exponential-type Orlicz spaces generated by $\exp\{|x|^p\} - 1$ be characterized in a way that facilitates probabilistic inequalities?
• RQ2 What is the optimal form of a Bernstein-type inequality for weighted sums of independent random variables in these Orlicz spaces?
• RQ3 Can the new norm characterization lead to tighter tail probability bounds than classical Bernstein inequalities?

Key findings

• A new equivalent norm characterization is established for Orlicz spaces generated by $\Psi_p(x) = \exp\{|x|^p\} - 1$, enabling precise analysis of moment and tail behavior.
• The paper derives a Bernstein-type inequality that holds for weighted sums of independent random variables in the $\Psi_p$-Orlicz space for $p \geq 1$.
• The resulting inequality improves upon classical Bernstein bounds by incorporating the specific growth structure of the $\Psi_p$ function.
• The method enables sharper concentration bounds for sums of heavy-tailed random variables compared to standard sub-Gaussian or sub-Exponential assumptions.
• The characterization allows for a unified treatment of moment and tail behavior across different $p \geq 1$ regimes.