Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
Using the inequality from question 19, prove the inequality $$(4) \quad b(M) \leq e n b^{\prime}(M).$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $Z$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ that follows the distribution $\mu_s$. Calculate $\mathbf{P}(k \mid Z)$ for $k \in \mathbb{N}^*$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $s \geqslant 2$ be an integer. Let $Z_n^{(s)}$ be the gcd of $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. Using the results of questions 18, 19, and 20a, deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.

We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. Show that there exists $\omega > 0$ such that $|\lambda^m - \lambda| \geqslant \omega$ for all integer $m \geqslant 2$.

We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. We consider the series $H \in O_2$ from part E. Show that the series $H$ satisfies $\hat{H} \prec \frac{1}{\omega} \hat{F} \circ (I + \hat{H})$.

Given a real $t > 0$, we set, following the notations of part $\mathbf{C}$,
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $\theta$, we set
$$h ( t , \theta ) = e ^ { - i m _ { t } \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) }$$
Let $\theta \in \mathbf { R }$ and $t \in \mathbf { R } _ { + } ^ { * }$. We consider, for all $k \in \mathbf { N } ^ { * }$, a random variable $Z _ { k }$ following the distribution $\mathcal { G } \left( 1 - e ^ { - k t } \right)$, and we set $Y _ { k } = k \left( Z _ { k } - \mathrm { E } \left( Z _ { k } \right) \right)$. Prove that
$$h ( t , \theta ) = \lim _ { n \rightarrow + \infty } \prod _ { k = 1 } ^ { n } \Phi _ { Y _ { k } } ( \theta )$$
Deduce, using in particular question $21 \triangleright$, the inequality
$$\left| h ( t , \theta ) - e ^ { - \frac { \sigma _ { t } ^ { 2 } \theta ^ { 2 } } { 2 } } \right| \leq K ^ { 3 / 4 } | \theta | ^ { 3 } S _ { 3,3 / 4 } ( t ) + K \theta ^ { 4 } S _ { 4,1 } ( t )$$

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$. We set $$\forall x \in ] - 1,1 [ , \quad g ( x ) = \sum _ { k = 0 } ^ { + \infty } x ^ { k }.$$ Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $] - 1,1 [$ and that $$\forall j \in \mathbb { N } , \quad \forall x \in ] - 1,1 [ , \quad g ^ { ( j ) } ( x ) = \frac { j ! } { ( 1 - x ) ^ { j + 1 } }.$$

Deduce that $$\int_{-\pi}^{\pi} e^{-i\frac{\theta^2}{6t^2}} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta = O(t^{3/2}) \text{ when } t \text{ tends to } 0^+.$$

In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$.
$\mathbf{25}$ ▷ After justifying that $E = E_s \oplus E_i \oplus E_n$, show that $$E_s = \left\{ X \in E \mid \lim_{t \rightarrow +\infty} e^{tA} X = 0 \right\}.$$

Taking $t = \frac { \pi } { \sqrt { 6 n } }$ in the formula
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta$$
conclude that
$$p _ { n } = O \left( \frac { \exp \left( \pi \sqrt { \frac { 2 n } { 3 } } \right) } { n } \right) \quad \text { when } n \text { tends to } + \infty$$

We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$. We consider the random column matrices $C = \left( \begin{array} { c } c _ { 1 } \\ \vdots \\ c _ { n } \end{array} \right)$ and $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$.
Deduce $\mathbb { P } \left( \left( C , C ^ { \prime } \right) \text { is linearly dependent} \right)$.

We set $A_2 = \operatorname{diag}(9, 5, 4)$. Justify the existence of a random vector whose covariance matrix is $A_2$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) = 0$. Show that there exists a vector $z \in E$ such that $\omega ( x , z ) \neq 0$ and $\omega ( y , z ) \neq 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Prove the following lemma: For all non-zero vectors $x$ and $y$ of $E$, there exists a composition $\gamma$ of at most two symplectic transvections of $E$ such that $\gamma ( x ) = y$.

Write in Python a function \texttt{modifie\_matrice(p, A)} that takes as arguments a probability $p$ and a numpy array representing a matrix $A \in \mathcal { V } _ { n , n }$. This function modifies the array A according to the following procedure: for every natural integer $k$, the matrix $A _ { k + 1 }$ is constructed from the matrix $A _ { k }$ by keeping each coefficient of $A _ { k }$ equal to $-1$ and by changing to $-1$ with probability $p$ each coefficient of $A _ { k }$ equal to $1$.

For any real $\alpha > 0$, consider $J _ { \alpha } = 2 \ln ( 2 ) - \ln \left( 1 + \alpha ^ { 2 } \right) - 2 \alpha \arctan \left( \frac { 1 } { \alpha } \right)$. Show that there exists $\gamma > 0$ such that, for all $\alpha \in ] 0 , \gamma [$, $J _ { \alpha } > 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $e _ { 1 } \in E$ be a non-zero vector. Justify the existence of $f _ { 1 } \in E$, not collinear with $e _ { 1 }$, such that $\omega \left( e _ { 1 } , f _ { 1 } \right) = 1$.

Using the function \texttt{modifie\_matrice}, write in Python a function \texttt{nb\_tours(p, n)} that takes as arguments a probability $p$ and the order $n$ of the matrices $A _ { k }$ and returns the smallest integer $k$ such that $A _ { k } = - A _ { 0 }$, starting from the matrix $A _ { 0 }$ (the real matrix of order $n$ whose coefficients are all equal to 1).

Give an example of an integer $K \in \mathbb{N}^{\star}$ for which the event in question Q33 occurs with the functions $f_n$ defined by $$\left\{\begin{array}{l} f_0 = 0 \\ f_n(x) = \ln\left(1 + \sin\left(\frac{x}{n}\right)\right) \quad \forall n \in \mathbb{N}^{\star}, \forall x \in [0,1]. \end{array}\right.$$

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1 \in E$ be a non-zero vector. Why does there exist a composition $\delta _ { 1 }$ of at most two symplectic transvections of $E$ such that $\delta _ { 1 } \left( u \left( e _ { 1 } \right) \right) = e _ { 1 }$?

Write in Python a function \texttt{moyenne\_tours(p, n, nbe)} that takes as arguments a probability $p$, the order $n$ of the matrices $A _ { k }$ and an integer \texttt{nbe} and that returns the average, over \texttt{nbe} trials performed, of the number of steps necessary to go from $A _ { 0 }$ to $- A _ { 0 }$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1, f_1 \in E$ with $\omega(e_1, f_1) = 1$, and let $\delta_1$ be a composition of at most two symplectic transvections such that $\delta_1(u(e_1)) = e_1$. Let $\tilde { f } _ { 1 }$ denote the vector $\delta _ { 1 } \left( u \left( f _ { 1 } \right) \right)$. Show that there exists a composition $\delta _ { 2 }$ of at most two symplectic transvections of $E$ such that
$$\left\{ \begin{array} { l } \delta _ { 2 } \left( e _ { 1 } \right) = e _ { 1 } \\ \delta _ { 2 } \left( \tilde { f } _ { 1 } \right) = f _ { 1 } \end{array} \right.$$
One may adapt the proof of the preceding lemma.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $P = \operatorname { Vect } \left( e _ { 1 } , f _ { 1 } \right)$ where $\omega(e_1, f_1) = 1$. Let $\delta = \delta_2 \circ \delta_1$ be a composition of at most four symplectic transvections satisfying $\delta(u(e_1)) = e_1$ and $\delta(u(f_1)) = f_1$. Set $v = \delta \circ u$. Show that $P$ is stable under $v$ and determine $v _ { P }$, the endomorphism induced by $v$ on $P$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $P = \operatorname { Vect} ( e_1, f_1 )$ with $\omega(e_1,f_1)=1$, and $v = \delta \circ u$ where $u \in \operatorname{Symp}_{\omega}(E)$ and $\delta$ is a composition of symplectic transvections with $v|_P = \mathrm{id}_P$. Show that $P ^ { \omega }$ is stable under $v$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $P = \operatorname{Vect}(e_1, f_1)$ with $\omega(e_1,f_1)=1$, and $v = \delta \circ u$ where $u \in \operatorname{Symp}_{\omega}(E)$ and $\delta$ is a composition of symplectic transvections with $v|_P = \mathrm{id}_P$. Show that the restriction $\omega _ { P ^ { \omega } }$ of $\omega$ to $P ^ { \omega } \times P ^ { \omega }$ equips $P ^ { \omega }$ with a symplectic space structure and that the endomorphism $v _ { P ^ { \omega } }$ induced by $v$ on $P ^ { \omega }$ is a symplectic endomorphism.