I read proof of Cramer's theorem from the book : 'Large Deviations and Applications' by A.Dembo and Ofer Zeitouni. I have a doubt in the proof of the lower bound.
Cramer's theorem(Lower bound) : Let $ X_1, X_2 , \ldots $ be i.i.d random variables with law $\mu$ . Let $S_n = \sum_{i = 1} ^{n} X_i $ , and let $\mu_n$ be the law of $S_n / n$. Let $\Lambda$ be the cumulant generating function of $X_1$ , and let $\Lambda^{*}$ be the corresponding rate function.We then have $$ \liminf\limits_{n \rightarrow \infty } \mu_n((-\delta,\delta)) \geq \inf_{\lambda \in \mathbb{R}} \Lambda(\lambda) = -\Lambda^{*}(0)$$
I have read the proof of the above under the assumptions that $\mu$ has bounded support , and would like to prove it without this assumption. Without any assumptions on the support of $\mu$ ,I have been able to prove that for all $M$ sufficiently large
$$ \liminf\limits_{n \rightarrow \infty } \mu_n((-\delta,\delta)) \geq \inf_{\lambda \in \mathbb{R}} \Lambda_M(\lambda)$$ where $ \Lambda_M(\lambda) = log \int_{-M}^{M} exp(\lambda x) d\mu(x)$
TO SHOW :$$ \liminf\limits_{n \rightarrow \infty } \mu_n((-\delta,\delta)) \geq \inf_{\lambda \in \mathbb{R}} \Lambda(\lambda)$$
We have that $\inf_{\lambda \in \mathbb{R}} \Lambda_M(\lambda)$ increases( in $M$ ) , but I do not know if it increases to $\inf_{\lambda \in \mathbb{R}} \Lambda(\lambda)$.