Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
Calculate the moment generating function of the random variable $S _ { n }$.

Show that, for any real $\xi$, there exists a real sequence $\left(c_{p}(\xi)\right)_{p \in \mathbb{N}}$ such that $$\forall x \in \mathbb{R}, \quad \exp\left(-x^{2}\right) \cos(2\pi \xi x) = \sum_{p=0}^{+\infty} c_{p}(\xi) \exp\left(-x^{2}\right) x^{2p}$$

We consider the family of matrices $B = \left[ b _ { i , j } \right] _ { 1 \leq i , j \leq n } \in \mathcal { M } _ { n } ( \mathbb { R } )$ satisfying the following three properties (called $M$-matrices):
$$\forall i \in \{ 1 , \ldots , n \} , \left\{ \begin{array} { l } b _ { i , i } > 0 \\ b _ { i , j } \leq 0 \text { for all } j \neq i \\ \sum _ { j = 1 } ^ { n } b _ { i , j } > 0 \end{array} \right.$$
Show that if $B$ is an $M$-matrix, then we have
(a) $B$ is invertible
(b) If $F = {}^{ t } \left( f _ { 1 } , \ldots , f _ { n } \right)$ has all positive coordinates, then $B ^ { - 1 } F$ also,
(c) all coefficients of $B ^ { - 1 }$ are positive.

Deduce that, for any real $\xi$, $\int_{-\infty}^{+\infty} \exp\left(-x^{2}\right) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \sqrt{\pi} \exp\left(-\pi^{2} \xi^{2}\right)$.

We set $\sigma^{\prime} = \frac{1}{2\pi\sigma}$. Show that there exists a real $\mu$ such that $\mathcal{F}\left(g_{\sigma}\right) = \mu g_{\sigma^{\prime}}$. The value of $\mu$ need not be made explicit.

Show that the function $\left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \times \mathbb{R} \rightarrow \mathbb{R} \\ & (t, x) \mapsto g_{\sqrt{\sigma^{2}+2t}}(x) \end{aligned}\right.$ satisfies conditions i and iii, where:

• [i.] the diffusion equation: $\forall(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \frac{\partial f}{\partial t}(t, x) = \frac{\partial^{2} f}{\partial x^{2}}(t, x)$;
• [iii.] the boundary condition: $\forall x \in \mathbb{R},\ \lim_{t \rightarrow 0^{+}} f(t, x) = g_{\sigma}(x)$.

Let $f$ be a function satisfying the diffusion equation, the three domination conditions, and the boundary condition $\lim_{t\to 0^+} f(t,x) = g_\sigma(x)$. Justify that, for any real $t > 0$ and any real $\xi$, the function $x \mapsto f(t, x) \exp(-2\mathrm{i}\pi \xi x)$ is integrable on $\mathbb{R}$.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that $\varphi$ is continuous on $\mathbb { R }$ and of class $C ^ { \infty }$ on $\mathbb { R } \backslash \{ 1 \}$.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Calculate $\lim _ { \substack { x \rightarrow 1 \\ x < 1 } } \varphi ^ { \prime } ( x )$ and demonstrate that $\varphi$ is of class $C ^ { 1 }$ on $\mathbb { R }$.

Let $A$ be a circulant matrix. Give a polynomial $P \in \mathbb{C}[X]$ such that $A = P(M_n)$.

Deduce the existence of a real $\nu_{\sigma}$ such that, for any real $\xi$ and any real $t > 0$, $$\hat{f}(t, \xi) = \nu_{\sigma} \exp\left(-2\pi^{2}\left(\sigma^{2}+2t\right) \xi^{2}\right)$$

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 $E'$. 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
$$\mathbb{P}(X \in C) = \frac{1}{2} \mathbb{P}(X' \in C_{+1}) + \frac{1}{2} \mathbb{P}(X' \in C_{-1})$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$. Show that $f$ attains a maximum at some point $x_0 \in \bar{U}$.

We assume $c(x) = 0$ for all $x \in [0,1]$. Let $u$ be the solution of problem (1) and $u_0, \ldots, u_n$ solutions of system (2) with $c = 0$. Define: $$\hat { B } _ { n + 1 } u ( X ) = \sum _ { k = 0 } ^ { n } u _ { k } \binom { n + 1 } { k } X ^ { k } ( 1 - X ) ^ { n + 1 - k }$$
Show that for all $n \in \mathbb { N } ^ { * }$ and all $x \in ]0,1[$ we have:
$$\left( \hat { B } _ { n + 1 } u \right) ^ { \prime \prime } ( x ) = - \frac { n } { n + 1 } \sum _ { \ell = 0 } ^ { n - 1 } f \left( \frac { \ell + 1 } { n + 1 } \right) \binom { n - 1 } { \ell } x ^ { \ell } ( 1 - x ) ^ { n - 1 - \ell }$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.

We assume $c(x) = 0$ for all $x \in [0,1]$, $f \in \mathcal{C}([0,1],\mathbb{R})$ satisfies $|f(y)-f(z)| \leq K|y-z|^\alpha$ for some $\alpha \in ]0,1]$ and $K \geq 0$. Let $u$ be the solution of problem (1) and define $\hat{B}_{n+1}u$ as above. Let $n \in \mathbb { N } ^ { * }$ such that $n \geq 2$. We set $\chi _ { n + 1 } = \hat { B } _ { n + 1 } u - u$.
(a) Show that
$$\left\| \chi _ { n + 1 } ^ { \prime \prime } \right\| _ { \infty } \leq \left\| f - B _ { n - 1 } f \right\| _ { \infty } + \frac { 1 } { n + 1 } \| f \| _ { \infty } + K \frac { 1 } { ( n + 1 ) ^ { \alpha } }$$
(b) Show that for all $x \in [ 0,1 ]$ there exists $\xi \in [ 0,1 ]$ such that
$$\chi _ { n + 1 } ( x ) = - \frac { 1 } { 2 } x ( 1 - x ) \chi _ { n + 1 } ^ { \prime \prime } ( \xi )$$
Hint: for $x \in ]0,1[$ one may consider the function
$$h ( t ) = \chi _ { n + 1 } ( t ) - \frac { \chi _ { n + 1 } ( x ) } { x ( 1 - x ) } t ( 1 - t ) , \quad t \in [ 0,1 ]$$

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 $E'$. 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 $C_{t}$. Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$.
Deduce that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + \left\|(1 - \lambda)(Y_{\varepsilon_{n}} - X') + \lambda(Y_{-\varepsilon_{n}} - X')\right\|^{2}$$
then that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda) \|Y_{\varepsilon_{n}} - X'\|^{2} + \lambda \|Y_{-\varepsilon_{n}} - X'\|^{2}$$
Thus, we have shown the inequality
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda) d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Show that $g_\varepsilon$ is a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that $\forall x \in U, \Delta g_\varepsilon(x) > 0$.

Deduce that, for any strictly positive real $t$, $f(t, \cdot) = g_{\sqrt{\sigma^{2}+2t}}$.

Deduce that there exists a constant $M \geq 0$ such that for all $n \in \mathbb { N } ^ { * }$, we have
$$\left\| u - \hat { B } _ { n + 1 } u \right\| _ { \infty } \leq \frac { M } { n ^ { \alpha / 2 } }$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Deduce that $\forall x \in U, f(x) \leqslant \sup_{y \in \partial U} f(y)$.

We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$. We have shown the inequality
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda) d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$
Show that for all $\lambda$ in $[0, 1]$
$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = -1\right) \leqslant \exp\left(\frac{\lambda^{2}}{2}\right) \mathbb{E}\left(\left(\exp\left(\frac{1}{8} d(X', C_{-1})^{2}\right)\right)^{1-\lambda} \cdot \left(\exp\left(\frac{1}{8} d(X', C_{+1})^{2}\right)\right)^{\lambda}\right)$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f_1$ and $f_2$ be two functions continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. Show that if the functions $f_1$ and $f_2$ are equal on $\partial U$, then $f_1$ and $f_2$ are equal on $U$.

We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$.
Deduce that
$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = -1\right) \leqslant \exp\left(\frac{\lambda^{2}}{2}\right) \left(\mathbb{E}\left(\exp\left(\frac{1}{8} d(X', C_{-1})^{2}\right)\right)\right)^{1-\lambda} \cdot \left(\mathbb{E}\left(\exp\left(\frac{1}{8} d(X', C_{+1})^{2}\right)\right)\right)^{\lambda}$$

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$.
Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand in power series on $D(0,R)$. What can we deduce about the function $f$?