QIII.C.3
Central limit theorem
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We fix two real numbers $a$ and $b$ such that $a < b$. For all integers $n \geqslant 1$ such that $a + \sqrt{n\lambda} > 0$, we define $$I_{n} = \{k \in \mathbb{N} \mid n\lambda + a\sqrt{n\lambda} \leqslant k \leqslant n\lambda + b\sqrt{n\lambda}\}$$ For $k \in \mathbb{Z}$, we define $x_{k,n} = \frac{k - n\lambda}{\sqrt{n\lambda}}$. We consider the function $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$.
For all $k \in I_{n}$, we denote $y_{k,n} = \left(1 - \frac{x_{k,n}}{k}\sqrt{n\lambda}\right)^{k} \exp\left(x_{k,n}\sqrt{n\lambda}\right)$.
Let $\varepsilon > 0$. Prove the existence of an integer $N(\varepsilon)$ such that, for all $n \geqslant N(\varepsilon)$ and all $k \in I_{n}$, the following inequalities are satisfied:
a) $\frac{1-\varepsilon}{\sqrt{2\pi}} \frac{1}{\sqrt{n\lambda}} y_{k,n} \leqslant \mathrm{e}^{-n\lambda} \frac{(n\lambda)^{k}}{k!} \leqslant \frac{1+\varepsilon}{\sqrt{2\pi}} \frac{1}{\sqrt{n\lambda}} y_{k,n}$;
We will use Stirling's formula $n! \underset{n \rightarrow +\infty}{\sim} \sqrt{2\pi n} \left(\frac{n}{\mathrm{e}}\right)^{n}$.
b) $(1-\varepsilon) f\left(x_{k,n}\right) \leqslant y_{k,n} \leqslant (1+\varepsilon) f\left(x_{k,n}\right)$.