Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ We denote $G_{X}(t) = \mathrm{E}\left(t^{X}\right) = \sum_{k=0}^{\infty} \mathrm{P}(X = k) t^{k}$ (generating series of the random variable $X$).
Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Determine $G_{X}(t)$.

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if $Y$ admits a finite expectation, that is, if the series $\sum n ^ { m } P ( Y = n )$ converges. We then call the moment of order $m$ of $Y$ the real number $$\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$$
Show that if $Y : \Omega \rightarrow \mathbb { N }$ is a random variable associated with a generating function $G _ { Y }$ of radius strictly greater than 1, then $Y$ admits a finite moment of all orders.

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We define $H_k(X) = X(X-1)\cdots(X-k+1)$ for $k \in \mathbb{N}^*$ and $H_0(X)=1$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if the series $\sum n ^ { m } P ( Y = n )$ converges, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
Let $Y : \Omega \rightarrow \mathbb { N }$ be a random variable admitting a finite moment of all orders.
IV.B.1) Show that the generating function $G _ { Y }$ is of class $C ^ { \infty }$ on $[ - 1,1 ]$.
IV.B.2) Express $G _ { Y } ^ { ( k ) } ( 1 )$ using the polynomials $H _ { k } ( X )$ and the random variable $Y$.
IV.B.3) Does the generating function $G _ { Y }$ necessarily have a radius of convergence strictly greater than 1? One may use the power series $\sum \mathrm { e } ^ { - \sqrt { n } } x ^ { n }$.

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We set for every integer $n \geqslant 0$, $B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$ where $S(n,k)$ is the number of partitions of $\llbracket 1,n \rrbracket$ into $k$ parts. Let $m$ be a strictly positive integer, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
We assume in this question that $Y$ follows the Poisson distribution with parameter 1.
IV.C.1) Show that for all $n \in \mathbb { N } , B _ { n } = \mathbb { E } \left( Y ^ { n } \right)$.
IV.C.2) Deduce that for every polynomial $Q ( X )$ with integer coefficients, the series $\sum _ { n = 0 } ^ { + \infty } \frac { Q ( n ) } { n ! }$ is convergent and its sum is of the form $N e$, where $N$ is an integer.

Let $X$ and $X^{\prime}$ be two random variables taking values in $\mathbb{N}$. Justify that $X \sim X^{\prime}$ if and only if $G_{X} = G_{X^{\prime}}$.

Let $X$ be a random variable taking values in $\mathbb{N}$ admitting a decomposition $X \sim Y + Z$, where $Y$ and $Z$ are independent random variables taking values in $\mathbb{N}$. What relation links $G_{X}, G_{Y}$ and $G_{Z}$?

Let $X$ be a random variable following the binomial distribution $\mathcal{B}(n, p)$ where $n \geqslant 1$ and $p \in ]0,1[$. Show that $X$ is decomposable if and only if $n \geqslant 2$.

Let $A(T) \in \mathbb{R}[T]$ be the polynomial: $A(T) = T^{4} + 2T + 1$.
a) Let $U(T)$ and $V(T)$ be two polynomials with non-negative real coefficients such that $U(T)V(T) = A(T)$. Show that one of the polynomials $U(T)$ or $V(T)$ is constant.
One may distinguish cases according to the values of the degrees of $U(T)$ and $V(T)$.
b) Deduce that there exists a decomposable random variable $X$ such that $X^{2}$ is not decomposable.
One may consider the polynomial $\frac{1}{4}A(T)$.

In this subsection, $n$ is a natural integer greater than or equal to 2 and $X$ is a random variable taking values in $\mathbb{N}$, defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and following the uniform distribution on $\llbracket 0, n-1 \rrbracket$: $$\mathbb{P}(X = k) = \frac{1}{n} \text{ if } k \in \llbracket 0, n-1 \rrbracket \text{ and } \mathbb{P}(X = k) = 0 \text{ otherwise}$$
We assume in this question that $n$ is not prime: there exist integers $a$ and $b$, greater than or equal to 2, such that $n = ab$.
a) Show that there exists a unique pair of integer-valued random variables $(Q, R)$ defined on $\Omega$ such that $$X = aQ + R \quad \text{and} \quad \forall \omega \in \Omega, R(\omega) \in \llbracket 0, a-1 \rrbracket$$ One may consider a Euclidean division.
b) Specify the distribution of $(Q, R)$, then the distributions of $Q$ and $R$.
c) Show that $X$ is decomposable. Deduce an expression of $G_{X}$ as a product of two non-constant polynomials that one will specify.

In this subsection, $n$ is a natural integer greater than or equal to 2 and $X$ is a random variable taking values in $\mathbb{N}$, defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and following the uniform distribution on $\llbracket 0, n-1 \rrbracket$: $$\mathbb{P}(X = k) = \frac{1}{n} \text{ if } k \in \llbracket 0, n-1 \rrbracket \text{ and } \mathbb{P}(X = k) = 0 \text{ otherwise}$$
We assume in this question that $n$ is a prime number and we establish that $X$ is not decomposable.
a) Show that it suffices to prove the following result: if $U$ and $V$ are monic polynomials in $\mathbb{R}[T]$ with coefficients in $\mathbb{R}_{+}$ such that $U(T)V(T) = 1 + T + \cdots + T^{n-1}$, then one of the two polynomials $U$ or $V$ is constant.
In what follows, we fix polynomials $U$ and $V$ in $\mathbb{R}[T]$ that are monic with coefficients in $\mathbb{R}_{+}$ such that $$U(T)V(T) = 1 + T + \cdots + T^{n-1}$$ We set $r = \deg U$ and $s = \deg V$ and we assume by contradiction that $r$ and $s$ are non-zero.
b) Show that $U(T) = T^{r} U\left(\frac{1}{T}\right)$ and $V(T) = T^{s} V\left(\frac{1}{T}\right)$.
We then denote $U(T) = 1 + u_{1}T + \cdots + u_{r-1}T^{r-1} + T^{r}$ and $V(T) = 1 + v_{1}T + \cdots + v_{s-1}T^{s-1} + T^{s}$ with $r \leqslant s$ (if necessary by exchanging the roles of $U$ and $V$).
c) Show that $\forall k \in \llbracket 1, r \rrbracket, u_{k}v_{k} = 0$.
d) Deduce that $\forall k \in \llbracket 1, r \rrbracket, u_{k} \in \{0,1\}$ and $v_{k} \in \{0,1\}$.
e) Conclude.
One may first show that all coefficients of $V$ take values in $\{0,1\}$.

Let $X$ and $Y$ be two random variables defined on $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in $\mathbb{N}$.
a) Show that if $A$ and $B$ are events in $\mathcal{A}$, and if $\bar{A}$ and $\bar{B}$ are their respective complementary events, then $$|\mathbb{P}(A) - \mathbb{P}(B)| \leqslant \mathbb{P}(A \cap \bar{B}) + \mathbb{P}(\bar{A} \cap B)$$
b) Deduce that, for all $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.

Let $\left(U_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of mutually independent random variables taking values in $\mathbb{N}$ such that the series of $\mathbb{P}\left(U_{i} \neq 0\right)$ is convergent.
a) Let $Z_{n} = \left\{\omega \in \Omega \mid \exists i \geqslant n, U_{i}(\omega) \neq 0\right\}$. Show that $(Z_{n})$ is a decreasing sequence of events and that $\lim_{n \rightarrow \infty} \mathbb{P}\left(Z_{n}\right) = 0$.
b) Deduce that the set $\left\{i \in \mathbb{N}^{*} \mid U_{i} \neq 0\right\}$ is almost surely finite.
c) We set $S_{n} = \sum_{i=1}^{n} U_{i}$ and $S = \sum_{i=1}^{\infty} U_{i}$. Justify that $S$ is defined almost surely. Show that $G_{S_{n}}$ converges uniformly to $G_{S}$ on $[-1,1]$.

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
Show that there exists a unique real sequence $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ such that, for all $k \in \mathbb{N}^{*}$ $$k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
For all $k \in \mathbb{N}^{*}$, show $$\left|\lambda_{k}\right| \mathbb{P}(X = 0) \leqslant \mathbb{P}(X = k) + \sum_{j=1}^{k-1} \left|\lambda_{j}\right| \mathbb{P}(X = k-j) \leqslant (1 - \mathbb{P}(X = 0))\left(1 + \sum_{j=1}^{k-1} \left|\lambda_{j}\right|\right)$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
For all $k \in \mathbb{N}^{*}$, show: $1 + \sum_{j=1}^{k} \left|\lambda_{j}\right| \leqslant \frac{1}{\mathbb{P}(X = 0)^{k}}$.

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
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 $H_X$ denotes its auxiliary power series.
For $t \in ]-\rho(X), \rho(X)[$, show $G_{X}^{\prime}(t) = H_{X}^{\prime}(t) G_{X}(t)$, then $G_{X}(t) = \exp\left(H_{X}(t)\right)$.

Let $X$ and $Y$ be two independent random variables, defined on the space $\Omega$ and taking values in $\mathbb{N}$, and let $H_{X}$ and $H_{Y}$ be their auxiliary power series. Show $H_{X+Y}(t) = H_{X}(t) + H_{Y}(t)$ for all real $t$ satisfying $|t| < \min(\rho(X), \rho(Y))$.

Let $X$ be a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and let $H_{X}$ be its auxiliary power series: $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$ We say that $X$ is $\lambda$-positive if $\lambda_{k} \geqslant 0$ for all $k \geqslant 1$. We assume in this subsection that $X$ is $\lambda$-positive.
For all $k \in \mathbb{N}^{*}$, show that $\lambda_{k} \leqslant \frac{\mathbb{P}(X = k)}{\mathbb{P}(X = 0)}$. Deduce that the series $\sum \lambda_{k}$ converges.

Let $X$ be a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and let $H_{X}$ be its auxiliary power series: $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$ We say that $X$ is $\lambda$-positive if $\lambda_{k} \geqslant 0$ for all $k \geqslant 1$. We assume in this subsection that $X$ is $\lambda$-positive.
Show that, for all $t \in [-1,1], G_{X}(t) = \exp\left(H_{X}(t)\right)$ and that $\sum_{k=1}^{\infty} \lambda_{k} = -\ln(\mathbb{P}(X = 0))$.

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 the random variable $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$.
a) For all $n \in \mathbb{N}^{*}$, show that $X_{n,1}$ is almost surely non-negative.
b) For all $n \in \mathbb{N}^{*}$, show that $\mathbb{P}\left(X_{n,1} = 0\right) > 0$.
c) Show that the random variables $X_{n,i}$ are almost surely taking values in $\mathbb{N}$.

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 the random variable $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$.
a) Show $\lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = 0\right) = 1$.
b) Deduce that, for all $i \in \mathbb{N}^{*}, \lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = i\right) = 0$.

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 the random variable $X_{n,1} + \cdots + X_{n,n}$ 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 $n H_{n} = H_{X}$.
b) Deduce, for all $n$ and $k$ in $\mathbb{N}^{*}$ $$k n \mathbb{P}\left(X_{n,1} = k\right) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}\left(X_{n,1} = k-j\right)$$

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 the random variable $X_{n,1} + \cdots + X_{n,n}$ follows the distribution of $X$.
For all $k \in \mathbb{N}^{*}$, show that the sequence $\left(n\mathbb{P}\left(X_{n,1} = k\right)\right)_{n \in \mathbb{N}^{*}}$ converges to $\lambda_{k}$. Deduce that $X$ is $\lambda$-positive.

a) Show the result announced at the beginning of subsection III.C, namely that the following three assertions are equivalent: (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 $X \sim \sum_{i=1}^{\infty} i X_{i}$.
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?