We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y$, and $\Sigma_X$ is a diagonal matrix.
Deduce that the eigenvalues of $\Sigma_Y$ are all positive.

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show successively that $Y ^ { 2 }$ and $| Y | ^ { 3 }$ have finite expectation, and that
$$\mathrm { E } \left( Y ^ { 2 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 1 / 2 } \quad \text { then } \quad \mathrm { E } \left( | Y | ^ { 3 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, let $S_N = \sum_{n=0}^N X_n a_n$. Express the expectation and variance of $S_{\phi(j+1)} - S_{\phi(j)}$ in terms of the terms of the sequence $(a_n)_{n \in \mathbb{N}}$.

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Express the expectation and variance of $S_{\phi(j+1)} - S_{\phi(j)}$ in terms of the terms of the sequence $(a_n)_{n \in \mathbb{N}}$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$.
We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$.
For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $P(r \mid X_n^{(i)})$ and show that $P(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} P\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}.$$

Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$. For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $\mathbf{P}(r \mid X_n^{(i)})$ and show that $\mathbf{P}(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} \mathbf{P}\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y = \left(\begin{array}{c} X_1 \\ \vdots \\ X_n \end{array}\right)$.
Prove that the total variance of $X$ is equal to that of $Y$.

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show, for all real $u$, the inequality
$$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$
Deduce that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$

Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\tau _ { n } = \operatorname { tr } \left( M _ { n } \right)$.
Calculate the expectation and the variance of the variable $\tau _ { n }$.

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix whose diagonal coefficients $\lambda_i$ are all positive. Prove the existence of a discrete random variable $Z$ with values in $\mathcal{M}_{n,1}(\mathbb{R})$ such that $\Sigma_Z = D$.

Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\delta _ { n } = \operatorname { det } \left( M _ { n } \right)$.
Calculate the expectation of the variable $\delta _ { n }$.

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define the events $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ For all $j \in \mathbb{N}$, prove that the events $B_{j,m}$, for $m$ ranging over $\llbracket \phi(j)+1, \phi(j+1) \rrbracket$, are pairwise disjoint and that we have the equality of events $$B_j = \bigcup_{\phi(j) < m \leqslant \phi(j+1)} B_{j,m}.$$

Let $A \in \mathcal{S}_n(\mathbb{R})$ be a symmetric matrix whose eigenvalues are positive. Prove the existence of a discrete random variable $Y$ with values in $\mathcal{M}_{n,1}(\mathbb{R})$ such that $\Sigma_Y = A$.

Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\delta _ { n } = \operatorname { det } \left( M _ { n } \right)$.
Prove that the variance of the variable $\delta _ { n }$ is equal to $n!$
One may expand $\delta _ { n }$ along a row and reason by induction.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), explain how to deduce the formula $$\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m}).$$

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define the events $$A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ Explain how to deduce the formula $\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m})$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $U = \left(\begin{array}{c} u_1 \\ \vdots \\ u_n \end{array}\right)$ in $\mathcal{M}_{n,1}(\mathbb{R})$. We define the discrete random variable $X = U^\top Y$.
Show that $X$ admits a variance and that $$\mathbb{V}(X) = U^\top \Sigma_Y U.$$

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
$$\left| h ( t , \theta ) - e ^ { - \frac { \sigma _ { t } ^ { 2 } \theta ^ { 2 } } { 2 } } \right| \leq K ^ { 3 / 4 } | \theta | ^ { 3 } S _ { 3,3 / 4 } ( t ) + K \theta ^ { 4 } S _ { 4,1 } ( t )$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. The objective is to show that $$\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1.$$ We denote by $r$ the rank of the covariance matrix of $Y$.
Handle the case where $r = n$.

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 $\ker \Sigma_Y$, and we have shown that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$ for all $j \in \llbracket 1, d \rrbracket$.
Conclude that $\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1$.

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

We set $A_2 = \operatorname{diag}(9, 5, 4)$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. In this question only, we assume that $Y$ is a random variable with values in $\mathcal{M}_{3,1}(\mathbb{R})$ such that $\Sigma_Y = A_2$. Determine the maximum of $q_Y$ on $C$, where $q_Y(U) = \mathbb{V}(U^\top Y)$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$ and $q_Y(U) = \mathbb{V}(U^\top Y)$.
In the general case, prove that the function $q_Y$ admits a maximum on $C$. Specify the value of this maximum as well as a vector $U_0 \in C$ such that $$\max_{U \in C} \mathbb{V}\left(U^\top Y\right) = \mathbb{V}\left(U_0^\top Y\right).$$

Express $X_n$ using the $U_i$, $1 \leq i \leq n$. Deduce the expectation $\mathrm{E}\left(X_n\right)$ and the variance $\mathrm{V}\left(X_n\right)$.

Let $X$ and $Y$ be two Bernoulli random variables, having parameters $\lambda \in ]0,1[$ and $\mu \in ]0,1[$, respectively. Calculate $d_{VT}\left(p_X, p_Y\right)$.