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 $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, and let $S_N = \sum_{n=0}^N X_n a_n$. With the events $$B_{j,m} = \left\{|S_m - S_{\phi(j)}| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \; |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\},$$ $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} |S_n - S_{\phi(j)}| > 2^{-j}\right\},$$ 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 $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ 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 $$\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m}).$$

In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$.
Calculate the probability of the event $M _ { 2 } \in \mathcal { N } _ { 2 }$.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), let $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$. Show that the function $$\left|\begin{array}{lll} \mathbb{R} & \rightarrow & \mathbb{R} \\ \alpha & \mapsto & 2^{\phi(j+1)-\phi(j)} \mathbb{P}\left(\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}\right) \end{array}\right.$$ takes values in $\mathbb{N}$ and is even.

In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$.
Calculate the probability of the event $M _ { 2 } \in \mathcal { G } \ell _ { 2 } ( \mathbb { R } )$.

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 $\mathcal { R }$.
Let $\left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right) \in \{ - 1,1 \} ^ { n }$. Calculate $\mathbb { P } \left( \left( c _ { 1 } = \varepsilon _ { 1 } \right) \cap \cdots \cap \left( c _ { n } = \varepsilon _ { n } \right) \right)$.

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 $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$.
Prove that, for all $\omega \in \Omega$, the family $\left( C ( \omega ) , C ^ { \prime } ( \omega ) \right)$ is linearly dependent if and only if there exists $\varepsilon \in \{ - 1,1 \}$ such that $C ^ { \prime } ( \omega ) = \varepsilon C ( \omega )$.

We recall that $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ are $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$, that $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ is the random matrix taking values in $\mathcal { V } _ { n , n }$ and we denote $$C _ { 1 } = \left( \begin{array} { c } m _ { 11 } \\ \vdots \\ m _ { n 1 } \end{array} \right) , \ldots , C _ { n } = \left( \begin{array} { c } m _ { 1 n } \\ \vdots \\ m _ { n n } \end{array} \right)$$ the random variables taking values in $\mathcal { V } _ { n , 1 }$ constituted by the columns of the matrix $M _ { n }$.
For all $j \in \llbracket 1 , n - 1 \rrbracket$, we denote by $R _ { j }$ the event $$\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent and } C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)$$ and $R _ { n }$ the event $$\left( C _ { 1 } , \ldots , C _ { n } \right) \text { is linearly independent.}$$
Show that $( R _ { 1 } , \ldots , R _ { n } )$ is a complete system of events.

With the notation of question 28, show that $$\mathbb { P } \left( M \notin \mathcal { G } \ell _ { n } ( \mathbb { R } ) \right) \leqslant \sum _ { j = 1 } ^ { n - 1 } \mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) .$$

Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$ (where $X(\Omega) = \{-1,1\}$, $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=\frac{1}{2}$). We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Prove that, for every real number $t$, $$\mathbb { E } ( \exp ( t \langle X \mid Y \rangle ) ) = \left( \operatorname { ch } \left( \frac { t } { n } \right) \right) ^ { n }$$

Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$. We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Deduce that, for every real number $t$, $$\mathbb { E } ( \exp ( t \langle X \mid Y \rangle ) ) \leqslant \exp \left( \frac { t ^ { 2 } } { 2 n } \right)$$

Show that the distribution of the random variable $X_n$ is given by $$\forall k \in \llbracket 0, n \rrbracket \quad P_n\left(X_n = k\right) = \frac{1}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$

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

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$. In the case where $n \geqslant 2$, define the events $$A _ { 1 } = \left\{ \left| R _ { 1 } \right| \geqslant 3 x \right\} \quad \text { and } \quad A _ { p } = \left\{ \max _ { 1 \leqslant i \leqslant p - 1 } \left| R _ { i } \right| < 3 x \right\} \cap \left\{ \left| R _ { p } \right| \geqslant 3 x \right\}$$ for $p \in \llbracket 2 , n \rrbracket$.
Express the event $A$ using the events $A _ { 1 } , A _ { 2 } , \ldots , A _ { n }$.

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$, and $A_1, \ldots, A_n$ as defined in Q13.
Show that we have $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \sum _ { p = 1 } ^ { n } \mathbb { P } \left( A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \right) .$$

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A_1, \ldots, A_n$ be as defined in Q13.
Justify that for all $p \in \llbracket 1 , n \rrbracket$, we have the inclusion $$A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \subset A _ { p } \cap \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} .$$

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$.
Deduce that $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} \right) .$$

Let $X$ and $Y$ be two independent random variables, taking values in $\mathbf{N}$, defined on the same probability space $(\Omega, \mathcal{A}, P)$. Prove the relation $$p_{X+Y} = p_X * p_Y$$

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$.
Conclude that $$\forall x > 0 , \quad \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\} \right) \leqslant 3 \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { p } \right| \geqslant x \right\} \right) .$$

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B_n$ is defined as in Q19.
Show that, for all $j \in \llbracket 0 , n \rrbracket$, $$\mathbb { P } \left( \left\{ T _ { n } = j \right\} \right) = \int _ { x _ { n , j } - 1 / \sqrt { n } } ^ { x _ { n , j } + 1 / \sqrt { n } } B _ { n } ( x ) \mathrm { d } x$$

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Consider a pair $( u , v )$ of real numbers such that $u < v$, and denote $$J _ { n } = \left\{ j \in \llbracket 0 , n \rrbracket \left\lvert \, \frac { n + u \sqrt { n } } { 2 } \leqslant j \leqslant \frac { n + v \sqrt { n } } { 2 } \right. \right\}$$
Justify that $$\mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \sum _ { j \in J _ { n } } \mathbb { P } \left( \left\{ T _ { n } = j \right\} \right)$$

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_{n}, \mathscr{P}(\mathfrak{D}_{n}))$ equipped with the uniform probability. We define a random variable $Y_{n}$ by $Y_{n}(\sigma) = \varepsilon(\sigma)$.
Explicitly state the distribution of $Y_{n}$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_{n}, \mathscr{P}(\mathfrak{D}_{n}))$ equipped with the uniform probability. We define a random variable $Y_{n}$ by $Y_{n}(\sigma) = \varepsilon(\sigma)$.
Calculate, for all $\varepsilon \in \{-1, 1\}$, $\lim_{n \rightarrow +\infty} \mathbb{P}\left(Y_{n} = \varepsilon\right)$.