Subgaussianity is hereditarily determined

Let $n$ be a positive integer, let $\mathit{X}=(X_1,\dots ,X_n)$ 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 ${\theta }_1{X}_1+\cdots +{\theta }_n{X}_n$ 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 $\{1,\dots ,n\}$.