We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Let $n \in \mathbb{N}$. Establish the equality $$P \left( S _ { 2 n } = 0 _ { 2 } \right) = \left( \frac { \binom { 2 n } { n } } { 4 ^ { n } } \right) ^ { 2 }$$

We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Give a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$. Determine the radius of convergence $R$ of the power series $\sum_{n \geqslant 1} a_n x^n$.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Justify that the function $S$ is of class $\mathcal{C}^\infty$ on $]-R, R[$ and, for every integer $n \in \mathbb{N}$, express $S^{(n)}(0)$ as a function of $n$.

Let $z \in D$. Show the convergence of the series $\sum _ { n \geq 1 } \frac { z ^ { n } } { n }$. Specify the value of its sum when $z \in ] - 1,1 [$. We denote
$$L ( z ) : = \sum _ { n = 1 } ^ { + \infty } \frac { z ^ { n } } { n }$$

Let $z \in D$. Show the convergence of the series $\sum _ { n \geq 1 } \frac { z ^ { n } } { n }$. Specify the value of its sum when $z \in ] - 1,1 [$. We denote
$$L ( z ) : = \sum _ { n = 1 } ^ { + \infty } \frac { z ^ { n } } { n }$$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent.

Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce the convergence of the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ for all $z$ in $D$. In what follows, we denote, for $z$ in $D$,
$$P ( z ) : = \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$$

Let $z \in D$. Show that the function $\Psi : t \mapsto ( 1 - t z ) e ^ { L ( t z ) }$ is constant on $[ 0,1 ]$, and deduce that
$$\exp ( L ( z ) ) = \frac { 1 } { 1 - z }$$

Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce that the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ is convergent for all $z$ in $D$.

Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$
Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.

Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } P \left( e ^ { - t + i \theta } \right) \mathrm { d } \theta$$
so that
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t + i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Calculate $b_2$, $b_4$ and $b_6$ then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.

Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$, where $F(x) = \sum_{n=1}^{+\infty} f_n(x)$.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), prove that if the event $B_j$ occurs, then there exist $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$ and $\alpha \in \{-1, +1\}$ such that the event $$\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}$$ also occurs. One may express $S_m - S_{\phi(j)}$ in terms of the two numbers $\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}$ with $\alpha = \pm 1$.

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 $$\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j).$$

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 $\mathbb{P}(B) = 0$.

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.

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 $\sum |S_{\phi(j+1)} - S_{\phi(j)}|$.

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.

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 $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ (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.

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'). Let $P_n \in \mathbb{R}_{K-1}[X]$ be a polynomial satisfying $P_n(x_\ell) = f_n(x_\ell)$ for all $\ell \in \llbracket 1, K \rrbracket$ (cf. question 7). Show that the event $$\left\{\begin{array}{l} \text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n (f_n - P_n)^{(k)} \text{ is uniformly convergent on } [0,1], \\ \text{the function } \sum_{n=0}^{+\infty} X_n (f_n - P_n) \text{ is of class } \mathcal{C}^K, \\ \text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n (f_n - P_n)\right)^{(k)} = \sum_{n=0}^{+\infty} X_n (f_n - P_n)^{(k)} \end{array}\right\}$$ has probability 1.

We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ (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 $$\left\{\begin{array}{l} \text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\ \text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\ \text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)} \end{array}\right\}$$ has probability 1.

Determine the domain of definition of $\sigma$ and justify that $\sigma$ is continuous on it, where $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.