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 the random variable $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$. a) Show $\lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = 0\right) = 1$. b) Deduce that, for all $i \in \mathbb{N}^{*}, \lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = i\right) = 0$.
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 the random variable $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$.
a) Show $\lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = 0\right) = 1$.
b) Deduce that, for all $i \in \mathbb{N}^{*}, \lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = i\right) = 0$.