A Maximal Large Deviation Inequality for Sub-Gaussian Variables

Abstract

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 \[P<(\max_{1\le i\le N}S_{i}>ε>) \le\exp<(-\frac{1}{N^2}\sum_{i=1}^{N}\frac{ε^{2}}{2σ_{i}^{2}}>), \] 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.

This paper has been withdrawn by the authors due to a crucial error in the last sentence of the proof of Theorem 1: "we can take the infimum of the r.h.s. over s, which yields (1)." This statement is only true if a single value of s yields the supremum of (ε_i s - ρ_i(s)) simultaneously for every i