Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote $$S_{k} = \sum_{i=1}^{k} U_{i}$$
Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by $\varphi(\lambda) = \ln\left(\mathbb{E}\left[e^{\lambda U_{1}}\right]\right)$. Establish that $$\forall \lambda \in \mathbb{R}, \quad \varphi(\lambda) \leqslant \frac{\lambda^{2}}{2}.$$

We introduce a uniformly distributed random variable $C : \Omega \rightarrow \mathcal{M}_{n}(\{-1,1\})$. For $\omega \in \Omega$, we denote by $C_{i,j}(\omega)$ the coefficients of the matrix $C(\omega)$.
Let $X = (x_{1}, \ldots, x_{n})$ and $Y = (y_{1}, \ldots, y_{n})$ be two arbitrary vectors in $\{-1,1\}^{n}$. Show that $\left(x_{i} y_{j} C_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ is a family of $n^{2}$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed.

We introduce a uniformly distributed random variable $C : \Omega \rightarrow \mathcal{M}_{n}(\{-1,1\})$. For $\omega \in \Omega$, we denote by $C_{i,j}(\omega)$ the coefficients of the matrix $C(\omega)$.
Show that for all $t \geqslant 0$, we have $$\mathbb{P}\left(M(C) \geqslant t n^{3/2}\right) \leqslant \exp\left(-\left(\frac{t^{2}}{2} - 2\ln 2\right)n\right).$$

Using the results of the previous questions (in particular the integral representation of $I_n$ from question 2, the bounds from question 3, and the Gaussian integral $\int_{-\infty}^{+\infty} e^{-x^{2}/2}\, dx = \sqrt{2\pi}$), deduce Stirling's formula: $$n! \underset{n \rightarrow \infty}{\sim} \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.$$

We fix $A \in \mathcal{M}_{n}(\{-1,1\})$ and denote $$m(A) := \min(S(A) \cap \mathbb{N}).$$
By drawing inspiration from the previous question and the methods developed in Parts II and III, show that we also have $$m(A) \leqslant \sqrt{2n \ln(2n)}.$$

We denote by $\mathbb{R}_{N}[X]$ the vector space of polynomials with real coefficients, of degree at most $N$. We define the set $A_{N}$ formed by $P \in \mathbb{R}_{N}[X]$, such that $P(-1) = P(1) = 1$, which furthermore satisfy $P(x) \geqslant 0$ for all $x$ in the interval $[-1,1]$. We define on $\mathbb{R}_{N}[X]$ a linear form $L$ by $L(P) = \int_{-1}^{1} P(x)\,dx$.
(a) Verify that $A_{N}$ is a convex subset of $\mathbb{R}_{N}[X]$.
(b) Show that the expression $$\|P\|_{1} = \int_{-1}^{1} |P(x)|\,dx$$ defines a norm on $\mathbb{R}_{N}[X]$.
(c) Show that $A_{N}$ is closed in the normed vector space $\left(\mathbb{R}_{N}[X], \|\cdot\|_{1}\right)$.

Let $a$ and $b$ be in $E$. Show the following relation and give a geometric interpretation:
$$\|a + b\|^{2} + \|a - b\|^{2} = 2(\|a\|^{2} + \|b\|^{2})$$

Show that $\mathcal{H}(U)$ is a vector subspace of $\mathcal{C}^2(U, \mathbb{R})$.

Justify that $\forall k \in \llbracket 1 , n \rrbracket , 0 \leqslant X ^ { k } \leqslant 1 + X ^ { n }$.

We denote by $\mathbb{R}_{N}[X]$ the vector space of polynomials with real coefficients, of degree at most $N$. We define the set $A_{N}$ formed by $P \in \mathbb{R}_{N}[X]$, such that $P(-1) = P(1) = 1$, which furthermore satisfy $P(x) \geqslant 0$ for all $x$ in the interval $[-1,1]$. We define on $\mathbb{R}_{N}[X]$ a linear form $L$ by $L(P) = \int_{-1}^{1} P(x)\,dx$. The infimum of $L$ on $A_N$ is denoted $a_N = \inf\{L(P) \mid P \in A_N\}$.
(a) Show that the infimum of $L$ on $A_{N}$ is attained.
In what follows, we denote by $B_{N}$ the set of $P \in A_{N}$ such that $L(P) = a_{N}$.
(b) Show that $B_{N}$ is a convex compact subset.
(c) Verify that $B_{N}$ contains an even polynomial.

Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.

Deduce that, if $X$ admits a moment of order $n \left( n \in \mathbb { N } ^ { * } \right)$, then $X$ admits moments of order $k$ for all $k \in \llbracket 1 , n - 1 \rrbracket$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
(a) What is the degree of $P_{j}$?
(b) Show that $P_{j}$ is an even or odd polynomial, depending on the value of $j$.
(c) Show that $P_{j}(1) = 1$ and $P_{j}(-1) = (-1)^{j}$.

We assume in this question that $U$ is path-connected. Determine the set of functions $f$ in $\mathcal{H}(U)$ such that $f^2$ also belongs to $\mathcal{H}(U)$.

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Justify that knowledge of the function $M _ { X }$ allows us to determine uniquely the sequence $\left( m _ { n } ( X ) \right) _ { n \in \mathbb { N } ^ { * } }$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
By means of integration by parts, show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is orthogonal in $\mathbb{R}_{n}[X]$.

Give a non-constant function belonging to $\mathcal{H}(U)$. Is the product of two harmonic functions a harmonic function?

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Using the results from the preamble, show that, for all $t \in ] - R _ { X } , R _ { X } [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation and that $M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right)$.

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.

We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Show that there exist $U \in \mathscr { M } _ { n , k } ( \mathbb { R } )$ and $\Lambda = \operatorname { diag } \left( \lambda _ { 1 } , \ldots , \lambda _ { k } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$ such that $S = U \Lambda U ^ { \mathrm { T } }$ with $\lambda _ { 1 } \geqslant \ldots \geqslant \lambda _ { k } > 0$ and $U ^ { \mathrm { T } } U = I _ { k }$.
(b) Show that $\operatorname { Im } ( S ) = \operatorname { Im } ( U )$ and that $U U ^ { \mathrm { T } }$ is the matrix of the orthogonal projection onto $\operatorname { Im } ( U )$ in $\mathbb { R } ^ { n }$.
(c) By setting $V = A ^ { \mathrm { T } } U D \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ where $D = \operatorname { diag } \left( 1 / \sqrt { \lambda _ { 1 } } , \ldots , 1 / \sqrt { \lambda _ { k } } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$, show that $V ^ { \mathrm { T } } V = I _ { k }$ and $\tilde { S } = V \Lambda V ^ { \mathrm { T } }$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that, for any real $\xi$, $\mathcal{F}\left(f^{\prime}\right)(\xi) = 2\mathrm{i}\pi\xi \mathcal{F}(f)(\xi)$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Justify that $g$ is of class $\mathcal{C}^2$ on $\mathbb{R}^{*+} \times \mathbb{R}$.

Show that, for any natural integer $p$, the function $x \mapsto x^{2p} \exp\left(-x^{2}\right)$ is integrable on $\mathbb{R}$.

We denote $M_{p} = \int_{-\infty}^{+\infty} x^{2p} \exp\left(-x^{2}\right) \mathrm{d}x$. For $p$ a natural integer, give a relation between $M_{p+1}$ and $M_{p}$ and deduce that, for all $p \in \mathbb{N}$, $$M_{p} = \frac{\sqrt{\pi}(2p)!}{2^{2p} p!}$$