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 $X$ 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?

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 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 $\{0,1\}$.

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?

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 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 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 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 that $X$ is $\lambda$-positive.
Let $(X_{i})$ be the sequence of random variables defined in II.C.3 (with parameters $\lambda_i$). Show that $X \sim \sum_{i=1}^{\infty} i X_{i}$.

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 $X$.
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 $X_{n,1} + \cdots + X_{n,n}$ 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 $X_{n,1} + \cdots + X_{n,n}$ 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 $\mathbb{N}^{*}$ $$kn\mathbb{P}\left(X_{n,1} = k\right) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}\left(X_{n,1} = k-j\right)$$