Let $X$ be an infinitely divisible random variable taking values in $\mathbb{N}$ and such that $\mathbb{P}(X = 0) > 0$. For all $n \in \mathbb{N}^{*}$, there exist $n$ independent random variables $X_{n,1}, \ldots, X_{n,n}$ with the same distribution such that $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$. a) For all $n \in \mathbb{N}^{*}$, show that $X_{n,1}$ is almost surely non-negative. b) For all $n \in \mathbb{N}^{*}$, show that $\mathbb{P}\left(X_{n,1} = 0\right) > 0$. c) Show that the random variables $X_{n,i}$ are almost surely taking values in $\mathbb{N}$.
Let $X$ be an infinitely divisible random variable taking values in $\mathbb{N}$ and such that $\mathbb{P}(X = 0) > 0$. For all $n \in \mathbb{N}^{*}$, there exist $n$ independent random variables $X_{n,1}, \ldots, X_{n,n}$ with the same distribution such that $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$.
a) For all $n \in \mathbb{N}^{*}$, show that $X_{n,1}$ is almost surely non-negative.
b) For all $n \in \mathbb{N}^{*}$, show that $\mathbb{P}\left(X_{n,1} = 0\right) > 0$.
c) Show that the random variables $X_{n,i}$ are almost surely taking values in $\mathbb{N}$.