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 $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Let $\mu$ be a strictly positive real number. Let $Y$ be a random variable following the Poisson distribution $\mathcal{P}(\mu)$ and such that $X$ and $Y$ are independent. Determine the distribution of $X + Y$.

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 $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$.

Let $n$ be a non-zero natural integer and let $X_{1}, \ldots, X_{n}$ be mutually independent random variables following Poisson distributions with respective parameters $\lambda_{1}, \ldots, \lambda_{n}$. Show that $X_{1} + \cdots + X_{n}$ follows a Poisson distribution with parameter $\lambda_{1} + \cdots + \lambda_{n}$.

Let $X$ be a Poisson random variable. Show that $X$ is infinitely divisible.

Let $r$ be a non-zero natural integer and let $X_{1}, \ldots, X_{r}$ be mutually independent Poisson random variables. Show that $\sum_{i=1}^{r} i X_{i}$ is an infinitely divisible random variable.

Let $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of non-negative real numbers. We assume that the series $\sum \lambda_{i}$ is convergent, and we denote $\lambda = \sum_{i=1}^{\infty} \lambda_{i}$.
Let $\left(X_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of independent random variables such that, for all $i$, $X_{i}$ follows a Poisson distribution with parameter $\lambda_{i}$. We agree that, if $\lambda_{i} = 0$, $X_{i}$ is the zero random variable.
a) Show that the series $\sum \mathbb{P}\left(X_{i} \neq 0\right)$ is convergent.
b) Show that the series $\sum_{i \geqslant 1} X_{i}$ is almost surely convergent and that its sum (defined almost surely) follows a Poisson distribution with parameter $\lambda$.
c) Show that the series $\sum_{i \geqslant 1} i X_{i}$ is almost surely convergent and that its sum $X = \sum_{i=1}^{\infty} i X_{i}$ defines an infinitely divisible random variable.

Let $X$ be a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and let $H_{X}$ be its auxiliary power series: $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$ We say that $X$ is $\lambda$-positive if $\lambda_{k} \geqslant 0$ for all $k \geqslant 1$. We assume in this subsection that $X$ is $\lambda$-positive.
Let $(X_{i})$ be the sequence of random variables defined in II.C.3 (with parameters $\lambda_i$). Show that $X \sim \sum_{i=1}^{\infty} i X_{i}$.

Let $n$ be a non-zero natural integer and let $X_{1}, \ldots, X_{n}$ be mutually independent random variables following Poisson distributions with respective parameters $\lambda_{1}, \ldots, \lambda_{n}$. Show that $X_{1} + \cdots + X_{n}$ follows a Poisson distribution with parameter $\lambda_{1} + \cdots + \lambda_{n}$.

Let $X$ be a Poisson random variable. Show that $X$ is infinitely divisible.

Let $r$ be a non-zero natural integer and let $X_{1}, \ldots, X_{r}$ be mutually independent Poisson random variables. Show that $\sum_{i=1}^{r} i X_{i}$ is an infinitely divisible random variable.

Let $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of non-negative real numbers. We assume that the series $\sum \lambda_{i}$ is convergent, and we denote $\lambda = \sum_{i=1}^{\infty} \lambda_{i}$.
Let $\left(X_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of independent random variables such that, for all $i$, $X_{i}$ follows a Poisson distribution with parameter $\lambda_{i}$. We agree that, if $\lambda_{i} = 0$, $X_{i}$ is the zero random variable.
a) Show that the series $\sum \mathbb{P}\left(X_{i} \neq 0\right)$ is convergent.
b) Show that the series $\sum_{i \geqslant 1} X_{i}$ is almost surely convergent and that its sum (defined almost surely) follows a Poisson distribution with parameter $\lambda$.
c) Show that the series $\sum_{i \geqslant 1} i X_{i}$ is almost surely convergent and that its sum $X = \sum_{i=1}^{\infty} i X_{i}$ defines an infinitely divisible random variable.