Given set $A = \{-4, 0, 1, 2, 8\}$, $B = \{x \mid x^3 = x\}$, then $A \cap B = $ ( )
A. $\{0, 1, 2\}$
B. $\{1, 2, 8\}$
C. $\{2, 8\}$
D. $\{0, 1\}$

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.

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.

Let $(A_n)_{n\in\mathbb{N}}$ be a sequence of events all with probability 1.
Show that $P\left(\bigcup_{n\in\mathbb{N}}\overline{A_n}\right)=0$. What can be deduced for $P\left(\bigcap_{n\in\mathbb{N}}A_n\right)$?

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 $\alpha+\beta=1$.

Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and $\pi$ the orthogonal projection onto $E'$. We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $C_{t} = \pi(C \cap H_{t})$ where $H_t = E' + te_n$. For $t$ in $\{-1, 1\}$, we denote by $Y_{t}$ the projection of $X'$ onto the non-empty closed convex set $C_{t}$.
Show that
$$\mathbb{P}(X \in C) = \frac{1}{2}\mathbb{P}(X' \in C_{+1}) + \frac{1}{2}\mathbb{P}(X' \in C_{-1})$$

We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, an outcome resulting from $n$ successive draws is modeled by the $n$-tuple indicating the color and number of the balls successively obtained. We denote by $\Omega_{n}$ the set of possible outcomes of these $n$ draws.
By examining the number of balls in the urn just before each draw, justify that, for $n \geqslant 1$, $$\operatorname{card}(\Omega_{n}) = (a_{0} + b_{0}) \times \cdots \times (a_{0} + b_{0} + s(n-1)) = s^{n} L_{n}\left(\frac{a_{0} + b_{0}}{s}\right).$$

We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, $\Omega_{n}$ denotes the set of possible outcomes of $n$ draws, and $X_{n}$ denotes the number of white balls present in the urn after $n$ draws. For $\omega \in \Omega_{n}$, $b(\omega)$ denotes the number of white balls present in the urn at the end of the $n$ draws modeled by $\omega$.
Show that, for all $n \in \mathbb{N}^{*}$ and all $k \in \mathbb{N}$, $$P(X_{n} = k) = \frac{\operatorname{card}(\{\omega \in \Omega_{n} ; b(\omega) = k\})}{\operatorname{card}(\Omega_{n})}.$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. If $n \in \mathbb{N}^*$, we denote $\{n \mid X\}$ the event ``$n$ divides $X$'' and $\{n \nmid X\}$ the complementary event.
Calculate $P(n \mid X)$ for $n \in \mathbb{N}^*$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. If $n \in \mathbb{N}^*$, we denote $\{n \mid X\}$ the event ``$n$ divides $X$'' and $\{n \nmid X\}$ the complementary event.
Let $r \geqslant 1$ be an integer. Show that $$P\left(\bigcap_{i=1}^{r}\left\{p_i \nmid X\right\}\right) = \prod_{i=1}^{r}\left(1 - p_i^{-s}\right).$$

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $E$. We call the distribution of the variable $X$ and denote by $\mu_X$ the map $$\begin{array}{rcl} \mu_X : & \mathscr{P}(E) & \rightarrow [0;1] \\ & A & \mapsto P(\{X \in A\}) \end{array}$$ where $\{X \in A\} = \{\omega \in \Omega \text{ such that } X(\omega) \in A\}$.
Verify that $\mu_X$ is a probability on $E$.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $E$. We call the law of the variable $X$ and we denote $\mu_X$ the application where $\{X \in A\} = \{\omega \in \Omega \text{ such that } X(\omega) \in A\}$. Verify that $\mu_X$ is a probability on $E$.

We recall that $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ are $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$, that $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ is the random matrix taking values in $\mathcal { V } _ { n , n }$ and we denote $$C _ { 1 } = \left( \begin{array} { c } m _ { 11 } \\ \vdots \\ m _ { n 1 } \end{array} \right) , \ldots , C _ { n } = \left( \begin{array} { c } m _ { 1 n } \\ \vdots \\ m _ { n n } \end{array} \right)$$ the random variables taking values in $\mathcal { V } _ { n , 1 }$ constituted by the columns of the matrix $M _ { n }$.
For all $j \in \llbracket 1 , n - 1 \rrbracket$, we denote by $R _ { j }$ the event $$\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent and } C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)$$ and $R _ { n }$ the event $$\left( C _ { 1 } , \ldots , C _ { n } \right) \text { is linearly independent.}$$
Show that $( R _ { 1 } , \ldots , R _ { n } )$ is a complete system of events.

With the notation of question 28, show that $$\mathbb { P } \left( M \notin \mathcal { G } \ell _ { n } ( \mathbb { R } ) \right) \leqslant \sum _ { j = 1 } ^ { n - 1 } \mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) .$$

With the notation of question 28, justify that, for all $j \in \llbracket 1 , n - 1 \rrbracket$, $$\mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) = \sum _ { \left( v _ { 1 } , \ldots , v _ { j } \right) \in \mathcal { V } _ { n , 1 } ^ { j } } \mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( v _ { 1 } , \ldots , v _ { j } \right) \right) \mathbb { P } \left( \left( C _ { 1 } = v _ { 1 } \right) \cap \cdots \cap \left( C _ { j } = v _ { j } \right) \right) .$$

With the notation of question 28, deduce that, for all $j \in \llbracket 1 , n - 1 \rrbracket$, $$\mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) \leqslant 2 ^ { j - n } .$$

Let $x, y, z$ be three distributions on $\mathbf{N}$. Prove the properties: $$\begin{gathered} 0 \leq d_{VT}(x, y) \leq 1 \\ d_{VT}(x, y) = 0 \Longleftrightarrow x = y \\ d_{VT}(y, x) = d_{VT}(x, y) \\ d_{VT}(x, z) \leq d_{VT}(x, y) + d_{VT}(y, z) \end{gathered}$$

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$. In the case where $n \geqslant 2$, define the events $$A _ { 1 } = \left\{ \left| R _ { 1 } \right| \geqslant 3 x \right\} \quad \text { and } \quad A _ { p } = \left\{ \max _ { 1 \leqslant i \leqslant p - 1 } \left| R _ { i } \right| < 3 x \right\} \cap \left\{ \left| R _ { p } \right| \geqslant 3 x \right\}$$ for $p \in \llbracket 2 , n \rrbracket$.
Express the event $A$ using the events $A _ { 1 } , A _ { 2 } , \ldots , A _ { n }$.

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$, and $A_1, \ldots, A_n$ as defined in Q13.
Show that we have $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \sum _ { p = 1 } ^ { n } \mathbb { P } \left( A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \right) .$$

Let $(x, y, u, v) \in \left(\mathcal{D}_{\mathbf{N}}\right)^4$. Show that, for all natural number $k$, $$|(x * y)(k) - (u * v)(k)| \leq \sum_{i+j=k} y(j)|x(i) - u(i)| + \sum_{i+j=k} u(i)|y(j) - v(j)|.$$

With the notation of the previous question, establish the inequality $$d_{VT}(x * y, u * v) \leq d_{VT}(x, u) + d_{VT}(y, v)$$

Let $G = ( S , A ) \in \Omega _ { n }$. Determine the probability $\mathbf { P } ( \{ G \} )$ of the elementary event $\{ G \}$ in terms of $p _ { n } , q _ { n } , N$ and $a = \operatorname { card } ( A )$. Then recover the fact that $\mathbf { P } \left( \Omega _ { n } \right) = 1$.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by: $$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$ Show that $$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. For $k \in \llbracket 0 , s _ { 0 } \rrbracket$, we denote: $$\Sigma _ { k } = \sum _ { \substack { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } \\ s _ { H \cap H ^ { \prime } } = k } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right)$$ Show that $\Sigma _ { 0 } \leq \left( \mathbf{E} \left( X _ { n } ^ { 0 } \right) \right) ^ { 2 }$.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. For $k \in \llbracket 0 , s _ { 0 } \rrbracket$, we denote: $$\Sigma _ { k } = \sum _ { \substack { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } \\ s _ { H \cap H ^ { \prime } } = k } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right)$$ Let $k \in \llbracket 1 , s _ { 0 } \rrbracket$; show that : $$\Sigma _ { k } \leq \sum _ { H \in \mathcal { C } _ { 0 } } \binom { s _ { 0 } } { k } \binom { n - s _ { 0 } } { s _ { 0 } - k } c _ { 0 } p _ { n } ^ { 2 a _ { 0 } } p _ { n } ^ { - \frac { k } { \omega _ { 0 } } }$$