Let $Y$ be a discrete random variable, independent of $X$, and $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a bounded function. Show that $$\mathbb{E}(\varphi(X, Y)) = \sum_{i=0}^{+\infty} p_i \mathbb{E}\left(\varphi\left(x_i, Y\right)\right)$$

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that for all $n \in \mathbb{N}$ and $x \in \mathbb{R}$, $$f_{n+1}(x) = g(x) + \sum_{i=0}^{+\infty} p_i f_n\left(x - x_i\right)$$

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The function $f$ is the pointwise limit of the sequence $f_n$. Show that the function $f$ satisfies the following equality on $\mathbb{R}$ $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$. Show that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, we have $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$.

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$, and such that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Deduce that if moreover the support of $h$ is included in $\mathbb{R}^+$, then for all $x \in \mathbb{R}$, $h(x) = 0$.

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 $\mathbb{R}^+$.

We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$, $$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$

We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Deduce that there exists a sequence of positive reals $\left(q_i\right)_{i \geqslant 0}$ such that for all $x \in \mathbb{R}$, $$f(x) = \sum_{i=0}^{+\infty} q_i g\left(x - y_i\right), \quad \text{and} \quad \sum_{i \in \mathbb{N},\, y_i \in [x-K, x]} q_i = \mathbb{E}(N(x-K, x)).$$

Let $a$ be a real number. Show $P(X \geqslant a) = 0 \quad \Longleftrightarrow \quad \forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) = 0$.

Let $a$ be a real number. Let $m$ and $n$ be in $\mathbb{N}$.
a) Show that $S_{m+n} - S_{m}$ and $S_{n}$ have the same distribution.
b) Let $b$ be a real number. Show $P\left(S_{m+n} \geqslant (n+m)b\right) \geqslant P\left(S_{n} \geqslant nb\right) P\left(S_{m} \geqslant mb\right)$.

Let $a$ be a real number. We suppose that $P(X \geqslant a) > 0$. Show that the sequence $\left(\frac{\ln\left(P\left(S_{n} \geqslant na\right)\right)}{n}\right)_{n \geqslant 1}$ is well-defined and admits a limit $\gamma_{a}$ that is negative or zero, satisfying $$\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\gamma_{a}}$$

Assume that $X$ is constant equal to $a \in \mathbb{R}$. Show that $X$ is infinitely divisible.

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 $\omega \in \Omega$.
Let $n \in \mathbb{N}^{*}$ and let $X_{1}, \ldots, X_{n}$ be independent random variables with the same distribution, such that $X_{1} + \cdots + X_{n}$ has the same distribution as $X$.
a) For all $i \in \llbracket 1, n \rrbracket$, show that $X_{i} \leqslant \frac{M}{n}$ almost surely, then $\left|X_{i}\right| \leqslant \frac{M}{n}$ almost surely.
b) Deduce that $\mathbb{V}(X) \leqslant \frac{M^{2}}{n}$, where $\mathbb{V}(X)$ denotes the variance of $X$.

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?

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 $k \in \llbracket 1 , n - 1 \rrbracket$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. Let $(A_{1}, \ldots, A_{m})$ be a complete system of events with non-zero probabilities. Show that
$$\mathbb{E}(X) = \sum_{i=1}^{m} \mathbb{P}(A_{i}) \cdot \mathbb{E}(X \mid A_{i})$$

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. Let $(A_{1}, \ldots, A_{m})$ be a complete system of events with non-zero probabilities. Show that
$$\mathbb{E}(X) = \sum_{i=1}^{m} \mathbb{P}(A_{i}) \cdot \mathbb{E}(X \mid A_{i})$$

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative real $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that
$$\mathbb{E}(X^{2}) = 2 \int_{0}^{+\infty} t \mathbb{P}(|X| \geqslant t) \, dt$$
You may denote $X^{2}(\Omega) = \{y_{1}, \ldots, y_{n}\}$ with $0 \leqslant y_{1} < y_{2} < \cdots < y_{n}$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that
$$\mathbb{E}(X^{2}) = 2 \int_{0}^{+\infty} t \mathbb{P}(|X| \geqslant t) \, dt$$
You may denote $X^{2}(\Omega) = \{y_{1}, \ldots, y_{n}\}$ with $0 \leqslant y_{1} < y_{2} < \cdots < y_{n}$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative real $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that the second moment of $X$ is less than or equal to $\frac{a}{b}$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that the second moment of $X$ is less than or equal to $\frac{a}{b}$.

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 $\zeta$.