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?
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?