Subgaussianity is hereditarily determined

Abstract:

Let $n$ be a positive integer, let $\boldsymbol {X}=(X_1,\dots ,X_n)$ be a random vector in $\mathbb {R}^n$ with bounded entries, and let $(\theta _1,\dots ,\theta _n)$ be a vector in $\mathbb {R}^n$. We show that the subgaussian behavior of the random variable $\theta _1 X_1+\dots +\theta _n X_n$ is essentially determined by the subgaussian behavior of the random variables $\sum _{i\in H} \theta _i X_i$ where $H$ is a random subset of $\{1,\dots ,n\}$.