Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a. Show that for all $i \in \mathbb{N}^*$ and all integer $k \geqslant i$, the limit of the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ depends only on $i$ and not on $k$. We denote this limit $\mu_\infty(x_i)$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a, with limits $\mu_\infty(x_i)$ as defined in 12b. Show that the application $$\begin{aligned} \psi : \mathbb{N}^* &\longrightarrow \mathbb{N}^* \\ k &\longmapsto \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{aligned}$$ is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.

Let $p \in \llbracket 1, d \rrbracket$. We define a relation on the set of bases of a subspace $V$ of dimension $p$ of $E$ by: $e$ and $e^{\prime}$ are in relation if $\operatorname{det}_e(e^{\prime}) > 0$ where $\operatorname{det}_e(e^{\prime})$ is the determinant of $e^{\prime}$ in the basis $e$. We admit that this relation is an equivalence relation on the set of bases of $V$ for which there exist exactly two equivalence classes called orientations of $V$. An oriented subspace is a pair $(V, C)$ where $C$ is an orientation of $V$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$.
(a) Show that if $e$ and $e^{\prime}$ are two free families of cardinal $p$ of $E$ then $\Omega_p(e)$ and $\Omega_p(e^{\prime})$ are collinear if and only if $\operatorname{Vect}(e) = \operatorname{Vect}(e^{\prime})$.
(b) Show that for all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, there exists a unique $\Psi(V, C) \in \mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$. Show that there exists a unique series $h \in O_1$ such that $h \circ f = I$, and that $(h)_1 = 1/\lambda$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Let $p \in \mathbb{R}^d$, set $$K_p := \{x \in K : p \cdot x \leqslant p \cdot y, \forall y \in K\}.$$ Show that $K_p$ is non-empty, convex and closed and that $\operatorname{Ext}(K_p) \subset \operatorname{Ext}(K)$.

We assume $n = 2m$ and denote by $J \in \mathcal { M } _ { 2m } ( \mathbb { R } )$ the matrix defined in blocks by
$$J = \left( \begin{array} { c c } 0 & - I _ { m } \\ I _ { m } & 0 \end{array} \right)$$
and $j$ the endomorphism of $\mathbb { R } ^ { n }$ canonically associated with $J$. Show that the map $b _ { s } : \left\lvert \, \begin{array} { c c l } \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( x , y ) & \mapsto & \langle x , j ( y ) \rangle \end{array} \right.$ is a symplectic form on $\mathbb { R } ^ { n }$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $f \in \mathcal{C}^{1}$ with real values. We assume that the set $C(f)$ of points in $]-\pi, \pi[$ where the function $f^{\prime}$ vanishes is finite. We denote by $\ell$ the cardinality of $C(f)$ and, if $\ell \geq 1$, we denote by $t_{1} < \cdots < t_{\ell}$ the elements of $C(f)$. We set $t_{0} = -\pi$ and $t_{\ell+1} = \pi$. For $0 \leq j \leq \ell$, let $\psi_{j}$ be the function from $\mathbf{R}$ to $\{0,1\}$ equal to 1 on $\left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$ and to 0 on $\mathbf{R} \backslash \left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$.
If $y \in \mathbf{R}$, show that the set $f^{-1}(\{y\}) \cap [-\pi, \pi[$ is finite with cardinality bounded by $\ell+1$; we denote this cardinality by $N(y)$. If $y \in \mathbf{R}$, express $N(y)$ in terms of $\psi_{0}(y), \ldots, \psi_{\ell}(y)$. Deduce the inequality $$V(f) \leq 2\max\{N(y); y \in \mathbf{R}\}\|f\|_{\infty}$$

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 $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
(a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathscr{A}_p(E, \mathbb{R})$.
(b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathscr{A}_p(E, \mathbb{R})$.

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$. Show that there exists a unique series $g \in O_1$ such that $f \circ g = I$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
(a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathcal{A}_p(E, \mathbb{R})$.
(b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathcal{A}_p(E, \mathbb{R})$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $\operatorname{Ext}(K)$ is non-empty (one may reduce to the case where $0 \in K$ and reason on the dimension of $K$).

Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $\lambda , \mu$ be real eigenvalues of $u$, and let $E _ { \lambda } ( u ) , E _ { \mu } ( u )$ be the associated eigenspaces. Show that, if $\lambda \mu \neq 1$, then the subspaces $E _ { \lambda } ( u )$ and $E _ { \mu } ( u )$ are $\omega$-orthogonal, that is:
$$\forall x \in E _ { \lambda } ( u ) , \quad \forall y \in E _ { \mu } ( u ) , \quad \omega ( x , y ) = 0$$

We denote by $\mathcal{R}_{n}$ the set of rational functions with no pole in $\mathbb{U}$ of the form $\frac{P}{Q}$ where $P$ and $Q$ are two elements of $\mathbf{C}_{n}[X]$.
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $$\forall z \in \mathbb{U}, \quad Q(z) \neq 0$$ For $t \in [-\pi, \pi]$, we set $$f(t) = F\left(e^{it}\right) = g(t) + ih(t) \quad \text{where} \quad (g(t), h(t)) \in \mathbf{R}^{2}$$ For $u \in [-\pi, \pi]$, we define a function $f_{u}$ from $[-\pi, \pi]$ to $\mathbf{R}$ by $$\forall t \in [-\pi, \pi], \quad f_{u}(t) = g(t)\cos(u) + h(t)\sin(u) = \operatorname{Re}\left(e^{-iu}F\left(e^{it}\right)\right) = \operatorname{Re}\left(e^{-iu}f(t)\right).$$
In this question, we fix $u \in [-\pi, \pi]$ and assume that $f_{u}$ is not constant. We also fix $y \in \mathbf{R}$. Using if necessary the expression of $f_{u}(t)$ as the real part of $e^{-iu}F\left(e^{it}\right)$ and Euler's formula for the real part, determine $S \in \mathbf{C}_{2n}[X]$ such that $$\forall t \in [-\pi, \pi], \quad f_{u}(t) = y \Longleftrightarrow S\left(e^{it}\right) = 0.$$ Deduce that the set $f_{u}^{-1}(\{y\}) \cap [-\pi, \pi[$ is finite with cardinality bounded by $2n$.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. We denote by $\mathbb{E}(X)$ the expectation of a real random variable $X$. Let $\mathscr{P}(E)$ be the set of subsets of $E$ and $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, P)$ and for every subset $A$ of $E$: $$|\mu_X(A) - \mu_Y(A)| \leqslant \mathbb{E}\left(|\mathbb{1}_{\{X \in A\}} - \mathbb{1}_{\{Y \in A\}}|\right)$$ and deduce that $\|\mu_X - \mu_Y\| \leqslant P(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathscr{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $h$ the unique series in $O_1$ such that $h \circ f = I$ and $g$ the unique series in $O_1$ such that $f \circ g = I$. Show that $g = h$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathcal{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)

Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. The standard symplectic form is $b_s(x,y) = \langle x, j(y) \rangle$ where $j$ is canonically associated with $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.

By observing that the function $|\cos|$ is $2\pi$-periodic, calculate, for $\omega \in \mathbf{R}$, the integral $$\int_{-\pi}^{\pi} |\cos(u - \omega)| \mathrm{d}u$$ Deduce that, if $(a,b) \in \mathbf{R}^{2}$, $$\int_{-\pi}^{\pi} |a\cos(u) + b\sin(u)| \mathrm{d}u = 4\sqrt{a^{2} + b^{2}}$$

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. We also define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \mapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$
Justify that the map $L$ is well defined.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined as in 15a.
Show that $P(X_n \neq X) \leqslant P(L \geqslant n)$ for all integer $n$ in $\mathbb{N}$.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined as in 15a.
Deduce that $\lim_{n \rightarrow +\infty} \|\mu_{X_n} - \mu_X\| = 0$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ defined on $(\Omega, \mathcal{B}, \mathbb{P})$ with real values and define the random vector $Y(\omega) = \left(\begin{array}{c} Y_1(\omega) \\ \vdots \\ Y_n(\omega) \end{array}\right)$. The covariance matrix $\Sigma_Y$ has general term $\sigma_{i,j} = \operatorname{cov}(Y_i, Y_j)$.
Verify that $\Sigma_Y$ is a symmetric matrix, that $$\Sigma_Y = \mathbb{E}\left((Y - \mathbb{E}(Y))(Y - \mathbb{E}(Y))^\top\right)$$ and that, if $U$ is a constant vector in $\mathcal{M}_{n,1}(\mathbb{R})$, then $$\Sigma_{Y+U} = \Sigma_Y.$$

Conclude that
$$\ln P \left( e ^ { - t } \right) = \frac { \pi ^ { 2 } } { 6 t } + \frac { \ln ( t ) } { 2 } - \frac { \ln ( 2 \pi ) } { 2 } + o ( 1 ) \quad \text { when } t \text { tends to } 0 ^ { + } .$$