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 $\theta \in [ - \pi , \pi ]$,
$$| h ( t , \theta ) | \leq e ^ { - \beta \left( \sigma _ { t } \theta \right) ^ { 2 } } \quad \text { or } \quad | h ( t , \theta ) | \leq e ^ { - \gamma \left( \sigma _ { t } | \theta | \right) ^ { 2 / 3 } }$$

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

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

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), deduce that $$\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j).$$

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 $$A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\}, \quad B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\}.$$ Deduce that $\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j)$.

We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$. It has been shown that $| U ( g ) ( x ) | \leqslant 4 \| g \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$ and $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$. Show that for all $x > 0 , | F ( x ) U ( g ) ( x ) | \leqslant \frac { 4 \| f \| \| g \| } { 1 + x }$.

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 $\ker \Sigma_Y$.
Prove that $$\forall j \in \llbracket 1, d \rrbracket, \quad \mathbb{V}\left(V_j^\top(Y - \mathbb{E}(Y))\right) = 0.$$

Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Conclude that
$$\int _ { - \pi \sigma _ { t } } ^ { \pi \sigma _ { t } } j ( t , u ) \mathrm { d } u \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } \sqrt { 2 \pi }$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and fix $\bar{x} \in C$ where $C := \{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\}$. Let $K$ be as defined in question 24. Deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is at most $k$.

With the notation of questions 23, 24 and 25, deduce that if $y \in \operatorname{Ext}(K)$ then the cardinality of $I_+(y) \cup I_-(y)$ is less than or equal to $k$.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $B_j$), denote by $B$ the event $\bigcap_{J \in \mathbb{N}} \bigcup_{j \geqslant J} B_j$. Show the equality $\mathbb{P}(B) = 0$.

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 $j \in \mathbb{N}$. Define $B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\}$. We denote by $B$ the event $\bigcap_{J \in \mathbb{N}} \bigcup_{j \geqslant J} B_j$. Show the equality $\mathbb{P}(B) = 0$.

We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$ where $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that for all $x \in ] 0,1] , | F ( x ) | \leqslant \| f \| \left( \mathrm { e } ^ { - 1 } - \ln ( x ) \right) ^ { 1 / 2 }$. One may use Question 19.

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 $\ker \Sigma_Y$.
Deduce that $\mathbb{P}\left(V_j^\top(Y - \mathbb{E}(Y)) = 0\right) = 1$.

We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$, and $G$ as defined in question (26).
Show that $$\hat{G}(r) \leqslant \left(\alpha_m + (1 + \alpha_m)\alpha_m^2 + \frac{\alpha_m(1 + \alpha_m)(1 + \alpha_m^2)}{1 - \alpha_m}\right) r \leqslant r$$ for all $r$ such that $$0 \leqslant r \leqslant \frac{1 - \alpha_m}{(1 + \alpha_m)(1 + \alpha_m^2)} \gamma_m r_0$$

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $B_j$, $B_{j,m}$), show that the event $$\left\{\exists J \in \mathbb{N}, \quad \forall j \geqslant J, \quad \forall n \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket, \quad |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\}$$ occurs with probability 1.

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 $j \in \mathbb{N}$. Show that the event $$\left\{\exists J \in \mathbb{N}, \quad \forall j \geqslant J, \quad \forall n \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}$$ occurs with probability 1.

We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$. Show the existence and calculate the values of the limits at $0$ and at $+ \infty$ of the function $t \mapsto F ( t ) U ( g ) ( t )$.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$), deduce that the event $$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$ also has probability 1. One may examine the series $\sum |S_{\phi(j+1)} - S_{\phi(j)}|$.

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 $j \in \mathbb{N}$. Deduce that the event $$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$ also has probability 1. One may examine the series $\sum \left|S_{\phi(j+1)} - S_{\phi(j)}\right|$.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$), conclude that the event $$\left\{\text{the series } \sum X_n a_n \text{ is convergent}\right\}$$ has probability 1.

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 $j \in \mathbb{N}$. Conclude that the event $$\left\{\text{the series } \sum X_n a_n \text{ is convergent}\right\}$$ has probability 1.

We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Show that one of the two hypotheses (H2') or (H2) (where (H2) states that for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent) implies the other.

We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Show that one of the two hypotheses (H2') or (H2) (studied in part II, where (H2) states that for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent) implies the other.

We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ (mutually independent), distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ satisfying hypotheses (H1) and (H2'). Show that the event $$\left\{\text{for all } \ell \in \llbracket 1, K \rrbracket, \text{ the series } \sum X_n f_n(x_\ell) \text{ is convergent}\right\}$$ has probability 1.