Centered and bounded implies subgaussian

Question

There's a result that any $B$-bounded centered random variable $X$ (i.e., $\mathbb{E}(X)=0$ and $|X|<B$) is sub-Gaussian with parameter $B \sqrt{2 \pi}$. Does it still true in $n$-dimensions? If yes can we calculate its parameter?

Answer 1

Suppose that $\mathbf X=(X_1,\dots,X_n)^T$ is bounded in the sense that for any $i$, $|X_i|\leq B_i$ for some $\mathbf B=(B_1,\dots,B_n)^T$, suppose also that $\mathbb E[\mathbf X]=\mathbf 0$.

Then for any unit vector $\mathbf u$, $X_{\mathbf u}=\langle \mathbf X,\mathbf u\rangle$ is such that $\mathbb E[X_{\mathbf u}]=0$ by linearity and using triangular inequality \begin{align*} |X_{\mathbf u}|&\leq \sum_{i} |u_i| |X_i|\\ &\leq \sum_i |u_i| B_i\\ &:= B_{\mathbf u} \end{align*}

Hence $X_{\mathbf u}$ is sub-Gaussian with parameter $B_{\mathbf u}\sqrt{2\pi}$.

Now I'm not sure about what the parameter is in your definition but I'm pretty sure it can be found easily at this point.