In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$. Show that the power series $\sum \lambda_{k} t^{k}$ has a radius of convergence $\rho(X)$ greater than or equal to $\mathbb{P}(X = 0)$. For all real $t$ in $]-\rho(X), \rho(X)[$, we set $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$
In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$.
Show that the power series $\sum \lambda_{k} t^{k}$ has a radius of convergence $\rho(X)$ greater than or equal to $\mathbb{P}(X = 0)$. For all real $t$ in $]-\rho(X), \rho(X)[$, we set
$$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$