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 $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ as well as $$D_n(\lambda) = \exp(\lambda n S_n - n \psi(\lambda))$$
(a) For $n \geqslant 2$ and $\lambda \geqslant 0$, show that $$E[(X_1 - m(\lambda))(X_2 - m(\lambda)) D_n(\lambda)] = 0$$
(b) Deduce that, for $n \geqslant 1$ and $\lambda \geqslant 0$, $$E[(S_n - m(\lambda))^2 D_n(\lambda)] \leqslant \frac{4}{n}.$$