In Cramér's theorem (large deviations), a rate function of random variable $Z$ is defined as $$ \Lambda^*(x) = \sup_{t \in \mathbb R}(t x - \Lambda(t)) $$ where $\Lambda(t)$ is the cumulant generating function of $Z$.
My question is if $Z$ is bounded from below and above, is the function $\Lambda^*(x)$
- continuous?
- differentiable?
- $\frac{d \Lambda^*}{d x}(x) < \infty$?
I have checked a few examples and the answers seem to be positive. But I don't know how to do this for general $Z$.
Any reference on the properties of this function would be greatly appreciated!