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 $X$.
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.

The three assertions to be shown equivalent are: (i) $X$ is infinitely divisible; (ii) $X$ is $\lambda$-positive; (iii) there exists a sequence $\left(X_{i}\right)_{i \geqslant 1}$ of independent Poisson variables, as in II.C.3, such that $X \sim \sum_{i=1}^{\infty} i X_{i}$.
a) Show the result announced at the beginning of subsection III.C, namely that assertions (i), (ii), and (iii) are equivalent for an infinitely divisible random variable $X$ taking values in $\mathbb{N}$ with $\mathbb{P}(X=0)>0$.
b) How to adapt this result to random variables taking values in $\mathbb{N}^{*}$?
c) Let $X$ be a random variable following the geometric distribution $\mathcal{G}(p)$, where $p \in ]0,1[$: $$\forall k \in \mathbb{N}^{*} \quad \mathbb{P}(X = k) = (1-p)^{k-1}p$$ Is the random variable $X$ infinitely divisible?