Conclude that there exists a unique bounded function with support in $\mathbb{R}^+$ solution of $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$

Show that the set $\Lambda_X := \bigcup_{n \in \mathbb{N}} \left\{y \in \mathbb{R} \mid \mathbb{P}\left(S_n = y\right) > 0\right\}$ is countable and included in $\mathbb{R}^+$.

Let $X$ be a bounded infinitely divisible random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We denote $M = \sup_{\Omega} |X|$, so that $|X(\omega)| \leqslant M$ for all $\omega \in \Omega$.
Conclude that $X$ is almost surely constant.

Is a binomial variable infinitely divisible?

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Give a necessary and sufficient condition on $x$ for $X$ to have a finite expectation. Express this expectation using $\zeta$.

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $k \in \mathbb{N}$, give a necessary and sufficient condition on $x$ for $X^{k}$ to have a finite expectation. Express this expectation using $\zeta$.

Deduce that, if $X$ admits a moment of order $n \left( n \in \mathbb { N } ^ { * } \right)$, then $X$ admits moments of order $k$ for all $k \in \llbracket 1 , n - 1 \rrbracket$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Let $j$ be an integer such that $1 \leqslant j \leqslant k$. Show that for all real $x , | x | ^ { j } \leqslant 1 + | x | ^ { k }$ and deduce that $X$ admits a moment of order $j$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$. What can we say about $X$ if $\alpha$ is zero ?

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Let $Y : \Omega \rightarrow \mathbb { R }$ be a random variable satisfying $Y ( \Omega ) = X ( \Omega )$ and, for all $n \in \mathbb { N }$, $$\mathbb { P } \left( Y = x _ { n } \right) = \frac { a _ { n } x _ { n } ^ { 2 k } } { \alpha }$$ Show that $\phi _ { Y }$ is of class $C ^ { 2 }$ on $\mathbb { R }$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Using the result of Q35, deduce that $X$ admits a moment of order $2 k + 2$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set. Let $k \in \mathbb { N } ^ { * }$. Deduce from the previous questions that if $\phi _ { X }$ is of class $C ^ { 2 k }$ on $\mathbb { R }$, then $X$ admits a moment of order $2 k$.

Show that if $p = \frac { 1 } { 2 }$, then $T$ does not admit an expectation.

Justify that for all $n \geq 0$, the random variables $\tilde { S } _ { n } , \tilde { I } _ { n }$ and $\tilde { R } _ { n }$ as well as the random variables $\Delta \tilde { S } _ { n } , \Delta \tilde { I } _ { n }$ and $\Delta \tilde { R } _ { n }$, have finite expectation.
Here $\Delta U_n = U_{n+1} - U_n$ and the random variables take values in $\{0, \ldots, M\}$.