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

Give an example of an integer $K \in \mathbb{N}^\star$ for which the event $$\left\{\begin{array}{l} \text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\ \text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\ \text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)} \end{array}\right\}$$ occurs with the functions $f_n$ defined by $$\left\{\begin{array}{l} f_0 = 0 \\ f_n(x) = \ln\left(1 + \sin\left(\frac{x}{n}\right)\right) \quad \forall n \in \mathbb{N}^\star, \forall x \in [0,1]. \end{array}\right.$$

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 $X(\omega)$. We also define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \mapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$
Justify that the map $L$ is well defined.