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 $X$. For all $k \in \mathbb{N}^{*}$, show that the sequence $\left(n\mathbb{P}\left(X_{n,1} = k\right)\right)_{n \in \mathbb{N}^{*}}$ converges to $\lambda_{k}$. Deduce that $X$ is $\lambda$-positive.
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 $X$.
For all $k \in \mathbb{N}^{*}$, show that the sequence $\left(n\mathbb{P}\left(X_{n,1} = k\right)\right)_{n \in \mathbb{N}^{*}}$ converges to $\lambda_{k}$. Deduce that $X$ is $\lambda$-positive.