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 $X$. Let $H_{X}$ be the auxiliary power series of $X$, as defined in question III.A.4, and let $\rho(X)$ be its radius of convergence. For all $n \in \mathbb{N}^{*}$, let $H_{n}$ be the auxiliary power series of $X_{n,1}$. a) For all $n \in \mathbb{N}^{*}$, show $nH_{n} = H_{X}$. b) Deduce, for all $n$ and $k$ in $\mathbb{N}^{*}$ $$kn\mathbb{P}\left(X_{n,1} = k\right) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}\left(X_{n,1} = k-j\right)$$
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 $X$.
Let $H_{X}$ be the auxiliary power series of $X$, as defined in question III.A.4, and let $\rho(X)$ be its radius of convergence. For all $n \in \mathbb{N}^{*}$, let $H_{n}$ be the auxiliary power series of $X_{n,1}$.
a) For all $n \in \mathbb{N}^{*}$, show $nH_{n} = H_{X}$.
b) Deduce, for all $n$ and $k$ in $\mathbb{N}^{*}$
$$kn\mathbb{P}\left(X_{n,1} = k\right) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}\left(X_{n,1} = k-j\right)$$