Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables with distribution $\mathcal{P}(\lambda)$, $S_{n} = X_{1} + \cdots + X_{n}$, and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.
Determine the limits, when $n \rightarrow +\infty$, of
$$\mathrm{P}\left(T_{n} \geqslant a\right), \quad \mathrm{P}\left(T_{n} = a\right), \quad \mathrm{P}\left(T_{n} > a\right) \quad \text{and} \quad \mathrm{P}\left(T_{n} \leqslant b\right)$$