Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ 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 $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ as well as $$D _ { n } ( \lambda ) = \exp \left( \lambda n S _ { n } - n \psi ( \lambda ) \right)$$
(a) For $n \geqslant 2$ and $\lambda \geqslant 0$, show that $$E \left[ \left( X _ { 1 } - m ( \lambda ) \right) \left( X _ { 2 } - m ( \lambda ) \right) D _ { n } ( \lambda ) \right] = 0$$
(b) Deduce that, for $n \geqslant 1$ and $\lambda \geqslant 0$, $$E \left[ \left( S _ { n } - m ( \lambda ) \right) ^ { 2 } D _ { n } ( \lambda ) \right] \leqslant \frac { 4 } { n }$$