A Simple Note on the Basic Properties of Subgaussian Random Variables

TL;DR

This note clarifies the fundamental properties of subgaussian random variables by introducing a $(\sigma, ho)$-subgaussian framework, aiding in the refinement of bounds and comparison of results in stochastic analysis.

Contribution

It introduces a $(\sigma, ho)$-subgaussian definition, extending the classical concept to better facilitate bounds refinement and result alignment.

Findings

Defines $(\sigma, ho)$-subgaussian variables with a new parameter $ ho$

Provides a basic description of subgaussianity properties

Facilitates refined bounds and comparisons in stochastic processes

Abstract

This note provides a basic description of subgaussianity, by defining $(\sigma, \rho)$-subgaussian random variables $X$ ($\sigma>0, \rho>0$) as those satisfying $\mathbb{E}(\exp(\lambda X))\leq \rho\exp(\frac{1}{2}\sigma^2\lambda^2)$ for any $\lambda\in\mathbb{R}$. The introduction of the parameter $\rho$ may be particularly useful for those seeking to refine bounds, or align results from different sources, in the analysis of stochastic processes and concentration inequalities.