A Maximal Large Deviation Inequality for Sub-Gaussian Variables

In this short note we prove a maximal concentration lemma for sub-Gaussian random variables stating that for independent sub-Gaussian random variables we have \[P<(\max_{1\le i\le N}S_{i}>\epsilon>) \le\exp<(-\frac{1}{N^2}\sum_{i=1}^{N}\frac{\epsilon^{2}}{2\sigma_{i}^{2}}>), \] where $S_i$ is the sum of $i$ zero mean independent sub-Gaussian random variables and $\sigma_i$ is the variance of the $i$th random variable.