Orlicz norm of random variable and variance In probability and statistics Orlicz norms are frequently used in concentration inequalities. For example, for Bernstein's inequality, we have versions for sub-exponential random variables using $\psi_1$-norm and for bounded random variables using variance. 

My first thought is that the $\psi_1$-norm version is more general, and includes the case of bounded random variables as a special case. However, an example of Bernoulli random variable with probability $1/n$ being $1$ and $1-1/n$ being $0$ suggests this is not true. For $n$ very large, the variance is roughly of the order $1/n$. However, its $\psi_1$-norm is roughly of the order  ${1}/{\log n}$. 

This suggests that $\psi_1$-norm (or similarly, $\psi_2$-norm) is actually very loose for very biased Bernoulli random variables. Is there any other notions like $\psi_1$-norm that can accommodate biased Bernoulli random variables?
The Kearns-Saul inequality states that if $X\sim Ber(p)$ then
$$ E[\exp(t(X-p))] = (1-p)e^{-tp}+pe^{t(1-p)} \le \exp\left(\frac{1-2p}{4\log((1-p)/p)}t^2\right).$$

The subgaussian constant $\frac{1-2p}{4\log((1-p)/p)}$ is optimal.
See http://ecp.ejpecp.org/article/view/2359 and especially the appendix in http://www.jmlr.org/papers/v16/berend15a.html for background and a slick proof.

The K-S inequality is a considerable improvement over Hoeffding and Bernstein for very small/large $p$.