Let $ X_1, X_2, \dots $ be a sequence of iid real random variables with finite logarithmic moment generating function $\Lambda$. Cramer's theorem says that, for $x > \operatorname E[X_1].$, $$\lim_{n \to \infty} \frac 1n \log P\left(\sum_{i=1}^n X_i = nx \right) = -\Lambda^*(x)$$ where $\Lambda^*$ is the Legendre transform of $\Lambda$.
Under which conditions does the limit hold uniformly in $x \in (a, b)$?
My end goal would be to prove the following: $$\lim_{n \to \infty} \chi_n(x) = -\partial_x\Lambda^*(x)$$ where $\chi_n(x) = \partial_x \log P(\sum_{i=1}^n X_i = x)$.