# Bound sub-Gaussian variance proxy by variance for $[-1,1]$-valued random variables

Consider a random variable $$X$$ taking values in $$[-1,1]$$ with mean $$0$$ and variance $$\sigma^2$$. A quantity $$V^2$$ is a sub-Gaussian variance proxy if $$\mathbb{E} \exp(\lambda X) \leq \exp(\lambda^2 V^2/2)$$ for all $$\lambda \in \mathbf{R}$$; we may take $$V^2$$ to be "the" sub-Gaussian variance proxy if it is the smallest $$V^2$$ satisfying the condition.

I have verified that, in general, there no constant $$C$$ for which $$V^2 \leq C \sigma^2$$; but is there a non-trivial bound for $$V^2$$ that depends on $$\sigma^2$$? Note that an absolute bound on $$V^2$$ follows because $$X$$ is bounded; I'm looking for a bound that depends on the variance $$\sigma^2$$.

Question. Is there a bound of the form

$$V^2 \leq f(\sigma^2) \quad \text{as } \sigma^2\to 0 \qquad \text{(i.e. for sufficiently small \sigma^2)}$$ for some function $$f$$ with $$f(x) \to 0$$ as $$x \to 0$$?

• How is $X$ zero mean if it takes values in $[0,1]$?
– nemo
May 29 '20 at 13:47
• Oops, typo there. Should have read $[-1,1]$. Thanks! May 29 '20 at 16:15