Let $X$ be a centred random variable with variance $\sigma^2$, and whose moment-generating function exists in an open neighbourhood of the origin.
Say that $X$ satisfies a 'Bernstein-type' MGF bound with parameter $b \in \left( 0, \infty \right)$ if there holds the inequality
$$\log \left(\mathbf{E} \left[ \exp \left( t \cdot X\right) \right] \right) \leq \frac{\frac{1}{2} \cdot \sigma^2 \cdot t^2}{1 - b \cdot | t |}, \qquad t \in \left( - \frac{1}{b}, \frac{1}{b} \right),$$
where it should be emphasised that the $\sigma^2$ appearing here is exactly the variance of $X$, rather than being a free parameter.
I have two quite basic questions about this:
- Is it clear that under the given assumptions, $X$ will necessarily satisfy some Bernstein-type MGF bound? This seems quite intuitive to me, but I have not really seen it discussed.
- Assuming that this is the case, it is natural to ask for the minimal (infimal, really) such $b$ for a given $X$; call this $b_\star \left( X \right)$. Is much known about this $b_\star \left( X \right)$?
- For instance, can it easily be related to some Orlicz-type norm?
- One main utility of this $b$ in many cases is that it gives some precise information on when the CLT effectively 'kicks in' for iid sums of $X$, so perhaps it will show up in this context. Though of course, the quantity is only relevant for the quite-restrictive class of $X$ with some exponential integrability, so perhaps it wouldn't have been as interesting classically in this context.
I remark in passing that if one can take $b = 0$, then such random variables are known as 'strictly' or 'strongly' sub-Gaussian, and a few interesting things are known in this case (see e.g. papers of Bobkov), which even have a Fourier-analytic character. I don't know the works well enough to intuit whether they imply anything about the case of $b > 0$.