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 $(\theta_1,\dots,\theta_n)$ 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 $\{1,\dots,n\}$.