We assume $m>1$. We study the Galton-Watson process with $Y_n$ the number of individuals in generation $n$ (starting from 1 individual).
Show that the probability that the sequence $(Y_n)_{n\in\mathbb{N}^*}$ takes any fixed value $k$ infinitely many times is zero.

We assume $m>1$. Let $\alpha$ be the probability of extinction and $\beta$ be the probability that the sequence $(Y_n)$ diverges to infinity.
Show that $\alpha+\beta=1$.

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 $\mathcal{P}(\lambda)$. Let $\mu$ be a strictly positive real number. Let $Y$ be a random variable following the Poisson distribution $\mathcal{P}(\mu)$ and such that $X$ and $Y$ are independent. Determine the distribution of $X + Y$.

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 $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of mutually independent random variables, with distribution $\mathcal{P}(\lambda)$, and $S_{n} = X_{1} + X_{2} + \cdots + X_{n}$. Determine the expectation and standard deviation of the random variables $S_{n}$ and $T_{n} = \frac{S_{n} - n\lambda}{\sqrt{n\lambda}}$.

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 $$\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\gamma_{a}}$$

The interval $I$ and the function $\varphi_{X}$ are defined as in question I.A.4. We suppose that $X$ satisfies $(C_{\tau})$ for some $\tau > 0$, is not almost surely constant, and $a > E(X)$.
Show that, for $n$ in $\mathbb{N}^{*}$ and $t$ in $I \cap \mathbb{R}^{+}$ $$E\left(\mathrm{e}^{tS_{n}}\right) = \left(\varphi_{X}(t)\right)^{n}, \quad P\left(S_{n} \geqslant na\right) \leqslant \frac{\varphi_{X}(t)^{n}}{\mathrm{e}^{nta}}$$

The interval $I$ and the function $\varphi_{X}$ are defined as in question I.A.4. We suppose that $X$ satisfies $(C_{\tau})$ for some $\tau > 0$, is not almost surely constant, and $a > E(X)$. We define the function $\chi : \begin{aligned} & I \rightarrow \mathbb{R} \\ & t \mapsto \ln\left(\varphi_{X}(t)\right) - ta \end{aligned}$
a) Show that the function $\chi$ is bounded below on $I \cap \mathbb{R}^{+}$. We denote by $\eta_{a}$ the infimum of $\chi$ on $I \cap \mathbb{R}^{+}$.
b) Give an equivalent of $\chi(t)$ as $t$ tends to 0. Deduce that $\eta_{a} < 0$.
c) Show $\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\eta_{a}}$. Deduce that $\gamma_{a} < 0$.
d) In each of the following two cases, determine the set of real numbers $a$ satisfying the conditions $P(X \geqslant a) > 0$ and $a > E(X)$; then, for $a$ satisfying these conditions, calculate $\eta_{a}$.
i. $X$ follows the Bernoulli distribution $\mathcal{B}(p)$ with $0 < p < 1$.
ii. $X$ follows the Poisson distribution $\mathcal{P}(\lambda)$ with $\lambda > 0$.

We suppose here that the infimum $\eta_{a}$ of the function $\chi$ on $I \cap \mathbb{R}^{+}$ is attained at a point $\sigma$ interior to $I \cap \mathbb{R}^{+}$. Let $t$ be a real number interior to $I$ and such that $t > \sigma$, $b$ be a real number such that $b > \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}$.
a) Calculate $\sum_{x \in X(\Omega)} \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$.
We then admit (if necessary by modifying $(\Omega, \mathcal{A}, P)$) that there exists a random variable $X^{\prime}$ on $(\Omega, \mathcal{A})$ such that $X^{\prime}(\Omega) = X(\Omega)$ and whose probability distribution is given by $$\forall x \in X(\Omega), \quad P\left(X^{\prime} = x\right) = \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$$ and that there exists a sequence $\left(X_{n}^{\prime}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables defined on $(\Omega, \mathcal{A}, P)$ all following the same distribution as $X^{\prime}$.
b) Show $$E\left(X^{\prime}\right) = \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}, \quad E\left(X^{\prime}\right) > a$$

We suppose here that the infimum $\eta_{a}$ of the function $\chi$ on $I \cap \mathbb{R}^{+}$ is attained at a point $\sigma$ interior to $I \cap \mathbb{R}^{+}$. Let $t$ be a real number interior to $I$ and such that $t > \sigma$, $b$ be a real number such that $b > \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}$. We admit that, if $n$ in $\mathbb{N}^{*}$ and if $f$ is a map from $X(\Omega)^{n}$ to $\mathbb{R}^{+}$, we have $$E\left(f\left(X_{1}^{\prime}, \ldots, X_{n}^{\prime}\right)\right) = \frac{E\left(f\left(X_{1}, \ldots, X_{n}\right) \mathrm{e}^{tS_{n}}\right)}{\varphi_{X}(t)^{n}}$$
a) For $n$ in $\mathbb{N}^{*}$, we set $S_{n}^{\prime} = \sum_{k=1}^{n} X_{k}^{\prime}$. Show $P\left(na \leqslant S_{n}^{\prime} \leqslant nb\right) \leqslant P\left(S_{n} \geqslant na\right) \frac{\mathrm{e}^{ntb}}{\varphi_{X}(t)^{n}}$.
$$\text{We may introduce the map } f : \left|\, \begin{array}{cl} X(\Omega)^{n} & \rightarrow \mathbb{R} \\ \left(x_{1}, \ldots, x_{n}\right) & \mapsto \begin{cases} 1 & \text{if } na \leqslant \sum_{i=1}^{n} x_{i} \leqslant nb \\ 0 & \text{otherwise} \end{cases} \end{array} \right.$$
b) Using questions I.B.2, II.B.2c and a) above, finally show that $\eta_{a} = \gamma_{a}$.

In this question you may use the results from II.B.2d.
a) Let $\alpha$ be in $]0, 1/2[$. For $n$ in $\mathbb{N}^{*}$, we set $$A_{n} = \left\{k \in \{0, \ldots, n\}, \left|k - \frac{n}{2}\right| \geqslant \alpha n\right\}, \quad U_{n} = \sum_{k \in A_{n}} \binom{n}{k}$$ Determine the limit of the sequence $\left(U_{n}^{1/n}\right)_{n \geqslant 1}$.
b) Let $\lambda$ be in $\mathbb{R}^{+*}$, $\alpha$ be in $]\lambda, +\infty[$. For $n$ in $\mathbb{N}^{*}$, we set $$T_{n} = \sum_{\substack{k \in \mathbb{N} \\ k \geqslant \alpha n}} \frac{n^{k} \lambda^{k}}{k!}$$ Determine the limit of the sequence $\left(T_{n}^{1/n}\right)_{n \geqslant 1}$.

Is a binomial variable infinitely divisible?

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 $[-1,1]$.

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.

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 $\mathbb{P}(X = 0) > 0$, 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 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 $X \sim \sum_{i=1}^{\infty} i X_{i}$.

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 $i \in \mathbb{N}^{*}, \lim_{n \rightarrow \infty} \mathbb{P}\left(X_{n,1} = i\right) = 0$.