Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$
$$\pi : \left\lvert \, \begin{aligned} E & \rightarrow E' \\ \sum_{i=1}^{n} x_{i} e_{i} & \mapsto \sum_{i=1}^{n-1} x_{i} e_{i} \end{aligned} \right.$$
We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $H_{t}$ the affine hyperplane $E' + te_{n}$ and $C_{t} = \pi(C \cap H_{t})$.
Show, for $x' \in E'$ and $t \in \{-1, 1\}$, that $x' \in C_{t} \Longleftrightarrow x' + te_{n} \in C$.

Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$
$$\pi : \left\lvert \, \begin{aligned} E & \rightarrow E' \\ \sum_{i=1}^{n} x_{i} e_{i} & \mapsto \sum_{i=1}^{n-1} x_{i} e_{i} \end{aligned} \right.$$
We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $H_{t}$ the affine hyperplane $E' + te_{n}$ and $C_{t} = \pi(C \cap H_{t})$.
Show that $C_{+1}$ and $C_{-1}$ are non-empty closed convex sets of $E'$.

For all $f, g \in \mathcal{E}$, we define $$(f \mid g) = \int_{-\infty}^{+\infty} f(y) g(y) \,\mathrm{d}y.$$
We define $\gamma_\lambda : \mathbf{R} \rightarrow \mathbf{R}$ by $\gamma_\lambda(y) = \exp\left(-y^2/\lambda\right)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$.
(a) Show that for all $f \in \mathcal{E}$, we have $(f \mid f) \geq 0$ with equality if and only if $f = 0$.
(b) Show that for all $x \in \mathbf{R}$, $\tau_x\left(\gamma_\lambda\right)$ belongs to $\mathcal{E}$.

Let $\lambda > 0$ be fixed. We consider the set $\mathcal{G}$ of functions $g$ that can be written in the form $g = \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda)$ where $n$ is a strictly positive integer and $\left((x_i, \alpha_i)\right)_{1 \leq i \leq n}$ is a family of elements of $\mathbf{R}^2$: $$\mathcal{G} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$
Show that $\mathcal{G}$ is a vector subspace of $\mathcal{E}$ and that it is the smallest vector subspace of $\mathcal{E}$ that contains all functions $\tau_x(\gamma_\lambda)$ for arbitrary $x \in \mathbf{R}$.

We use the notation $\mathcal{G}$, $\mathcal{H}$, $\gamma_{2\lambda}$, $\tau_x$, $C$, $D$ as defined previously.
(a) Let $n \in \mathbf{N}_*$ and $(x_i)_{1 \leq i \leq n}$ a family of real numbers such that for all $i, j \in \llbracket 1,n \rrbracket$ we have $x_i \neq x_j$ when $i \neq j$. Show that the function $\sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda})$ is zero if and only if $\alpha_i = 0$ for all $1 \leq i \leq n$ (Hint: One may proceed by induction on $n$).
(b) Deduce that there exists a unique linear map $D$ from $\mathcal{H}$ to $\mathcal{G}$ such that $D \circ C(g) = g$ for all $g \in \mathcal{G}$ and $C \circ D(h) = h$ for all $h \in \mathcal{H}$.
(c) Show that for all $h \in \mathcal{H}$, we have for all $x \in \mathbf{R}$ that $h(x) = \left(\tau_x(\gamma_\lambda) \mid D(h)\right)$.

For all $(h_1, h_2) \in \mathcal{H} \times \mathcal{H}$, we denote $(h_1 \mid h_2)_{\mathcal{H}} = c_\lambda \left(D(h_1) \mid D(h_2)\right)$ where $c_\lambda$ is introduced in question (11a).
(a) Verify that $(\mid)_{\mathcal{H}}$ defines an inner product on $\mathcal{H}$.
(b) Show that for all $x \in \mathbf{R}$ and $h \in \mathcal{H}$ we have $h(x) = \left(\tau_x(\gamma_{2\lambda}) \mid h\right)_{\mathcal{H}}$.
(c) Show that for all $h \in \mathcal{H}$ we have $$\|h\|_\infty \leq \|h\|_{\mathcal{H}}$$ where we have set $\|h\|_\infty = \sup_{x \in \mathbf{R}} |h(x)|$ and $\|h\|_{\mathcal{H}} = (h \mid h)_{\mathcal{H}}^{1/2}$.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. Justify that, for all $(f,g) \in E^2$, we have $$\langle T(f), g \rangle = \langle f, T(g) \rangle$$ One may use question 12.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. We admit that, $$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$ Deduce that $H = \{0\}$.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. Show that the family of vectors $(g_k)_{k \in \mathbb{N}^*}$ is orthonormal.

We are given a real $a > 0$. Let $E_4$ be the space of functions continuous on $[0,a]$, taking values in $\mathbb{R}$, of class $\mathcal{C}^1$ piecewise and furthermore satisfying $f(a) = 0$. Let $\varphi:[0,a] \rightarrow \mathbb{R}$ be of class $\mathcal{C}^1$ satisfying $\varphi(a) = 0$ and, for all $x \in [0,a]$, $\varphi'(x) < 0$. Determine an inner product on $E_4$ such that the function $(x,y) \mapsto \min(\varphi(x), \varphi(y))$ is a reproducing kernel on the pre-Hilbert space $E_4$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Let $x \in I$ and $V_x$ defined on $E$ by $V_x(f) = f(x)$. We set $$N(V_x) = \sup_{\|f\|=1} |f(x)|$$ Prove that $$N(V_x) = \sqrt{\langle k_x, k_x \rangle}.$$

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ We consider a function $A:[0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T: E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t)\,\mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. Justify that $T$ induces an isomorphism from $(\ker T)^\perp$ onto $\operatorname{Im} T$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We assume $r < n$ where $r$ is the rank of $\Sigma_Y$. We denote by $d = \dim \ker \Sigma_Y$ and we consider an orthonormal basis $(V_1, \ldots, V_d)$ of $\ker \Sigma_Y$.
Deduce that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is at most $k$.

With the notation of questions 23, 24 and 25, deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is less than or equal to $k$.

For all functions $f \in E$ and $g \in E$, we set $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that this defines an inner product on $E$.

For all integer $p \in \mathbb { N } ^ { * }$ and all $x > 0$, we set $P _ { p } ( x ) = x \mathrm { e } ^ { x } g _ { p } ^ { ( p ) } ( x )$, where $g _ { p } ( x ) = x ^ { p - 1 } \mathrm { e } ^ { - x }$. We recall that $P _ { p }$ is a polynomial function of degree $p$, that $P _ { p } \in E$, and that $P_p$ is an eigenvector of $U$ for the eigenvalue $1/p$. The inner product on $E$ is $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that the polynomials $P _ { p }$ are pairwise orthogonal in $E$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and let $\mu$ be an element of $\mathscr{M}(E)$. Show that if the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in the normed vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$, then $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x).$$

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x). \tag{1}$$ We also fix a real number $\varepsilon > 0$.
Show that there exists a finite subset $F_\varepsilon$ of $E$ and an integer $N_\varepsilon \geqslant 0$ such that $\mu(F_\varepsilon) > 1 - \varepsilon$ and for all integer $n \geqslant N_\varepsilon$ $$\sum_{x \in F_\varepsilon} |\mu_n(x) - \mu(x)| < \varepsilon.$$

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying condition (1). We also fix a real number $\varepsilon > 0$ and a finite subset $F_\varepsilon$ of $E$ and integer $N_\varepsilon \geqslant 0$ as in 10a.
Show that for every subset $A$ of $E$: $$|\mu_n(A) - \mu(A)| \leqslant |\mu_n(A \cap F_\varepsilon) - \mu(A \cap F_\varepsilon)| + \mu(E \backslash F_\varepsilon) + \mu_n(E \backslash F_\varepsilon)$$ and deduce that if $n \geqslant N_\varepsilon$, then $|\mu_n(A) - \mu(A)| < 4\varepsilon$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Deduce that the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ if and only if it satisfies condition $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x). \tag{1}$$

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 $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and let $\mu$ be an element of $\mathscr{M}(E)$. Show that if the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in the normed vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$, then $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x)$$

Let $E$ be an infinite countable 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 $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. We define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \longmapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$ Justify that the application $L$ is well defined.

We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote $$Q = \left\{(q_{ij})_{(i,j) \in I \times J} \in \mathbb{R}^{I \times J} \mid q_{ij} \geq 0 \text{ for all } (i,j) \in I \times J\right\}$$ and $$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$ Verify that $F(\alpha, \beta)$ is a convex set of the vector space $E = \mathbb{R}^{I \times J}$.

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Show that the map $\varphi$ defined on $\left( L ^ { 0 } ( \Omega ) \right) ^ { 2 }$ by $$\forall X , Y \in L ^ { 0 } ( \Omega ) , \quad \varphi ( X , Y ) = \mathbf { E } ( X Y )$$ is an inner product on $L ^ { 0 } ( \Omega )$.