Subgaussianity is hereditarily determined

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 $(θ_1,\dots,θ_n)$ be a vector in $\mathbb{R}^n$. We show that the subgaussian behavior of the random variable $θ_1 X_1+\dots +θ_n X_n$ is essentially determined by the subgaussian behavior of the random variables $\sum_{i\in H} θ_i X_i$ where $H$ is a random subset of $\{1,\dots,n\}$.

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