Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$:
$$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$
Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$. For all integers $n \geqslant 1$, determine the distribution of $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$.