grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2017 centrale-maths2__mp

30 maths questions

QI.A.1 Probability Generating Functions Uniqueness and characterization of distributions via PGF View
Let $X$ and $X^{\prime}$ be two random variables taking values in $\mathbb{N}$. Justify that $X \sim X^{\prime}$ if and only if $G_{X} = G_{X^{\prime}}$.
QI.A.2 Probability Generating Functions PGF of sum of independent variables View
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 $G_{Z}$?
QI.A.3 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $n \geqslant 2$.
QI.A.4 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $\frac{1}{4}A(T)$.
QI.B.1 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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.
QI.B.2 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $\{0,1\}$.
QII.A.1 Discrete Random Variables Independence Proofs for Discrete Random Variables View
Assume that $X$ is constant equal to $a \in \mathbb{R}$. Show that $X$ is infinitely divisible.
QII.A.2 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
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 $X$.
QII.A.3 Discrete Random Variables Existence of Expectation or Moments View
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.
QII.B.1 Discrete Random Variables Existence of Expectation or Moments View
Is a binomial variable infinitely divisible?
QII.B.2 Sum of Poisson processes View
Let $n$ be a non-zero natural integer and let $X_{1}, \ldots, X_{n}$ be mutually independent random variables following Poisson distributions with respective parameters $\lambda_{1}, \ldots, \lambda_{n}$. Show that $X_{1} + \cdots + X_{n}$ follows a Poisson distribution with parameter $\lambda_{1} + \cdots + \lambda_{n}$.
QII.B.3 Sum of Poisson processes View
Let $X$ be a Poisson random variable. Show that $X$ is infinitely divisible.
QII.B.4 Sum of Poisson processes View
Let $r$ be a non-zero natural integer and let $X_{1}, \ldots, X_{r}$ be mutually independent Poisson random variables. Show that $\sum_{i=1}^{r} i X_{i}$ is an infinitely divisible random variable.
QII.C.1 Probability Generating Functions Bounding probabilities or tail estimates via PGF View
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 $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.
QII.C.2 Probability Generating Functions Bounding probabilities or tail estimates via PGF View
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 $[-1,1]$.
QII.C.3 Sum of Poisson processes View
Let $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of non-negative real numbers. We assume that the series $\sum \lambda_{i}$ is convergent, and we denote $\lambda = \sum_{i=1}^{\infty} \lambda_{i}$.
Let $\left(X_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of independent random variables such that, for all $i$, $X_{i}$ follows a Poisson distribution with parameter $\lambda_{i}$. We agree that, if $\lambda_{i} = 0$, $X_{i}$ is the zero random variable.
a) Show that the series $\sum \mathbb{P}\left(X_{i} \neq 0\right)$ is convergent.
b) Show that the series $\sum_{i \geqslant 1} X_{i}$ is almost surely convergent and that its sum (defined almost surely) follows a Poisson distribution with parameter $\lambda$.
c) Show that the series $\sum_{i \geqslant 1} i X_{i}$ is almost surely convergent and that its sum $X = \sum_{i=1}^{\infty} i X_{i}$ defines an infinitely divisible random variable.
QIII.A.1 Probability Generating Functions Recursive or recurrence relation via PGF coefficients View
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 $k \in \mathbb{N}^{*}$ $$k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$$
QIII.A.2 Probability Generating Functions Recursive or recurrence relation via PGF coefficients View
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 $$\left|\lambda_{k}\right| \mathbb{P}(X = 0) \leqslant \mathbb{P}(X = k) + \sum_{j=1}^{k-1} \left|\lambda_{j}\right| \mathbb{P}(X = k-j) \leqslant (1 - \mathbb{P}(X = 0))\left(1 + \sum_{j=1}^{k-1} \left|\lambda_{j}\right|\right)$$
QIII.A.3 Probability Generating Functions Recursive or recurrence relation via PGF coefficients View
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: $1 + \sum_{j=1}^{k} \left|\lambda_{j}\right| \leqslant \frac{1}{\mathbb{P}(X = 0)^{k}}$.
QIII.A.4 Probability Generating Functions Radius of convergence and analytic properties of PGF View
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 $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$
QIII.A.5 Probability Generating Functions Radius of convergence and analytic properties of PGF View
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 $G_{X}(t) = \exp\left(H_{X}(t)\right)$.
QIII.A.6 Probability Generating Functions PGF of sum of independent variables View
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 $|t| < \min(\rho(X), \rho(Y))$.
QIII.B.1 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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.
QIII.B.2 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $\sum_{k=1}^{\infty} \lambda_{k} = -\ln(\mathbb{P}(X = 0))$.
QIII.B.3 Sum of Poisson processes View
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.
Let $(X_{i})$ be the sequence of random variables defined in II.C.3 (with parameters $\lambda_i$). Show that $X \sim \sum_{i=1}^{\infty} i X_{i}$.
QIII.C.1 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $\mathbb{N}$.
QIII.C.2 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $i \in \mathbb{N}^{*}, \lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = i\right) = 0$.
QIII.C.3 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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 $\mathbb{N}^{*}$ $$k n \mathbb{P}\left(X_{n,1} = k\right) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}\left(X_{n,1} = k-j\right)$$
QIII.C.4 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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.
QIII.C.5 Probability Generating Functions Infinite divisibility and decomposability via PGF View
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?