Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ as well as, for all $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ Show that for all $t \in \mathbb{R}$, we have $$\frac{1}{n} \log P[S_n \geqslant t] \leqslant \inf_{\lambda \geqslant 0} (\psi(\lambda) - \lambda t).$$
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$,
$$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$
We define
$$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$
as well as, for all $\lambda \in \mathbb{R}$,
$$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$
Show that for all $t \in \mathbb{R}$, we have
$$\frac{1}{n} \log P[S_n \geqslant t] \leqslant \inf_{\lambda \geqslant 0} (\psi(\lambda) - \lambda t).$$