We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$. We define $u_n$ and $u_n^{(r)}$ as above.
Let $n\in\mathbb{N}^*$ and $r$ a natural integer greater than or equal to 2. Show the relation $$u_n^{(r)}=\sum_{i=1}^{n-1}u_i u_{n-i}^{(r-1)}$$

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0. We define $u_n$, $u_n^{(r)}$, $U(s)=\sum_{n=1}^{+\infty}u_n s^n$ and $U_r(s)=\sum_{n=1}^{+\infty}u_n^{(r)}s^n$ for $s\in[-1,1]$.
Deduce that, for every strictly positive integer $r$, $U_r=U^r$ ($U^r$ denotes $U\times U\times\cdots\times U$ $r$ times).

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$. We define $u_n$, $u_n^{(r)}$, $U(s)$ and $U_r(s)$ as above, and $u$ is the probability that $(W_n)$ does not take the value $k$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ infinitely many times is zero.

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 $G_{X}(t)$.

Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$
Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Calculate the expectation $\mathrm{E}(X)$, the variance $V(X)$ and the standard deviation of $X$.

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 $G_{Z}$?

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 $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$

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

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real.
a) Show that the variable $X$ has finite expectation. We will denote by $m$ the expectation of $X$.
b) Apply, with appropriate justifications, the weak law of large numbers to the sequence of random variables $\left(X_{k}\right)$.

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real.
a) Show that the function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{t X}\right)$ is defined and continuous on the segment $[-\alpha, \alpha]$.
b) Show that the function $\Psi$ is differentiable on the interval $]-\alpha, \alpha[$ and determine its derivative function.

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$.
We consider the function $f_{\varepsilon}$ defined by $$f_{\varepsilon}:\left\{\begin{array}{l}[-\alpha, \alpha] \rightarrow \mathbb{R}^{+} \\ t \mapsto \mathrm{e}^{-(m+\varepsilon) t} \Psi(t)\end{array}\right.$$
a) Give the values of $f_{\varepsilon}(0)$ and $f_{\varepsilon}^{\prime}(0)$.
b) Deduce that there exists a real $t_{0}$ belonging to the interval $]0, \alpha[$ satisfying $0 < f_{\varepsilon}\left(t_{0}\right) < 1$.

We consider a general balanced urn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$. We denote by $g_{n}$ the generating function of $X_{n}$ (the number of white balls after $n$ draws).
Justify the equalities $$\begin{aligned} & g_{n}(t) = \frac{1}{\operatorname{card}(\Omega_{n})} P_{n}(t, 1) \\ & E(X_{n}) = \frac{1}{\operatorname{card}(\Omega_{n})} \frac{\partial P_{n}}{\partial u}(1,1) \end{aligned}$$

We assume in this question that $X ( \Omega )$ is a finite set of cardinality $r \in \mathbb { N } ^ { * }$. We denote $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ where the $x _ { i }$ are pairwise distinct, and, for all integer $k \in \llbracket 1 , r \rrbracket , a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all real $t , \phi _ { X } ( t ) = \sum _ { k = 1 } ^ { r } a _ { k } \mathrm { e } ^ { \mathrm { i } t x _ { k } }$.

Explicitly calculate the generating function $G_{X_1}$ of the random variable $X_1$, where $X_1$ follows a Poisson distribution with parameter $1/2$.

We assume in this question that $X ( \Omega )$ is a countable set. We denote $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ where the $x _ { n }$ are pairwise distinct. For all $n \in \mathbb { N }$, we set $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Show that $\phi _ { X }$ is defined on $\mathbb { R }$ and that, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { \mathrm { i } t x _ { n } }$.

Justify that $\forall t \in \mathbb{R}, G_{S_n}(t) = \left(G_{X_1}(t)\right)^n$.

Show that $\phi _ { X }$ is continuous on $\mathbb { R }$.

Let $a$ and $b$ be two real numbers and $Y = a X + b$. For all real $t$, express $\phi _ { Y } ( t )$ in terms of $\phi _ { X } , t , a$ and $b$.

Let $t \in \mathbb { R }$. Give an expression of $\phi _ { X } ( - t )$ in terms of $\phi _ { X } ( t )$. Deduce a necessary and sufficient condition on the image $\phi _ { X } ( \mathbb { R } )$ for the function $\phi _ { X }$ to be even.

Let $n \in \mathbb { N } ^ { * }$ and $\left. p \in \right] 0,1 [$. We assume that $X : \Omega \rightarrow \mathbb { R }$ follows a binomial distribution $\mathcal { B } ( n , p )$ and we denote $q = 1 - p$. Show that, for all $t \in \mathbb { R } , \phi _ { X } ( t ) = \left( q + p \mathrm { e } ^ { \mathrm { i } t } \right) ^ { n }$.

Let $p \in ] 0,1 [$. What is the characteristic function of a random variable following a geometric distribution with parameter $p$ ?

Let $\lambda > 0$. What is the characteristic function of a random variable following a Poisson distribution with parameter $\lambda$ ?

Let $X : \Omega \rightarrow \mathbb { R }$ and $Y : \Omega \rightarrow \mathbb { R }$ be two discrete random variables such that $\phi _ { X } = \phi _ { Y }$. Show that, for all $m \in \mathbb { R } , \mathbb { P } ( X = m ) = \mathbb { P } ( Y = m )$, in other words that $X$ and $Y$ have the same distribution.

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 $k$. Deduce that $\phi _ { X }$ is of class $C ^ { k }$ on $\mathbb { R }$ and give an expression of the $k$-th derivative of $\phi _ { X }$.