Show that there exists a real $\alpha > 0$ such that
$$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$
Using question $9 \triangleright$, deduce from this that there exist three real numbers $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $\left. t \in ] 0 , t _ { 0 } \right]$ and all $\theta \in [ - \pi , \pi ]$,
$$| h ( t , \theta ) | \leq e ^ { - \beta \left( \sigma _ { t } \theta \right) ^ { 2 } } \quad \text { or } \quad | h ( t , \theta ) | \leq e ^ { - \gamma \left( \sigma _ { t } | \theta | \right) ^ { 2 / 3 } }$$

Show that there exists a real $a > 0$ such that $$\forall \theta \in [-\pi,\pi], 1-\cos\theta \geq a\theta^2.$$ Deduce that there exist three reals $t_0 > 0$, $\beta > 0$ and $\gamma > 0$ such that, for all $t \in ]0,t_0]$ and all $\theta \in [-\pi,\pi]$, $$\left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\beta(t^{-3/2}\theta)^2} \quad \text{or} \quad \left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\gamma(t^{-3/2}|\theta|)^{2/3}}.$$

With the notation of question 28, deduce that $$\mathbb { P } \left( M \in \mathcal { G } \ell _ { n } ( \mathbb { R } ) \right) \geqslant \frac { 1 } { 2 ^ { n - 1 } } .$$

Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$
By applying Markov's inequality to a suitably chosen random variable, prove that $$\forall t \in \mathbb { R } ^ { + } , \quad \mathbb { P } ( Z \geqslant \lambda ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } - \lambda t \right)$$

Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$
Deduce that $$\mathbb { P } ( | Z | \geqslant \lambda ) \leqslant 2 \exp \left( - \frac { \lambda ^ { 2 } } { 2 \sigma ^ { 2 } } \right)$$

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 $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Prove that $$\mathbb { P } ( | \langle X \mid Y \rangle | \geqslant \varepsilon ) \leqslant 2 \exp \left( - \frac { \varepsilon ^ { 2 } n } { 2 } \right)$$

$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$.
Deduce from the previous questions that $$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) \leqslant N ( N - 1 ) \exp \left( - \frac { \varepsilon ^ { 2 } n } { 2 } \right) .$$

$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$.
We assume that $n \geqslant 4 \frac { \ln N } { \varepsilon ^ { 2 } }$. Prove that $$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) < 1 .$$

$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$.
Deduce that, for every natural integer $N$ less than or equal to $\exp \left( \frac { \varepsilon ^ { 2 } n } { 4 } \right)$, there exists a family of $N$ unit vectors of $\mathbb { R } ^ { n }$ whose coherence parameter is bounded by $\varepsilon$.

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. Let $S_N = \sum_{n=0}^N X_n a_n$ and let $A_j = \{|S_{\phi(j+1)} - S_{\phi(j)}| > 2^{-j}\}$. Using the sequence $(\phi(j))_{j \in \mathbb{N}}$ satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$, deduce the bound $\mathbb{P}(A_j) \leqslant 2^{-j}$.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined as in 15a.
Show that $P(X_n \neq X) \leqslant P(L \geqslant n)$ for all integer $n$ in $\mathbb{N}$.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined in 15a. Show that $\mathbf{P}(X_n \neq X) \leqslant \mathbf{P}(L \geqslant n)$ for all integer $n$ in $\mathbb{N}$.

Let $x > 0$. By writing that $\varphi ( t ) \leqslant \frac { t } { x } \varphi ( t )$ for all $t \geqslant x$, show that $$\int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t \leqslant \frac { \varphi ( x ) } { x }$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

Let $x > 0$. Using the study of a well-chosen function, show that $$\frac { x } { x ^ { 2 } + 1 } \varphi ( x ) \leqslant \int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

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 $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} \right) .$$

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 $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Deduce that: for all $t \geq 0$, for all $x \geq 0$ and for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \exp \left( x \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| \right) > \mathrm { e } ^ { t x } \right) \leq 2 \mathrm { e } ^ { - t x } \exp \left( \frac { x ^ { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } { 2 } \right) .$$ You may use Markov's inequality.

Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that: for all $t \geq 0$ and for all non-zero $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| > t \right) \leq 2 \exp \left( - \frac { t ^ { 2 } } { 2 \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } \right) .$$