A Simple Note on the Basic Properties of Subgaussian Random Variables

This note provides a basic description of subgaussianity, by defining $(\sigma ,\rho )$-subgaussian random variables $X$ ($\sigma >0,\rho >0$) as those satisfying $\mathbb{E}(\exp (\lambda X))\le \rho \exp (\frac{1}{2}{\sigma }^2{\lambda }^2)$ 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.