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'). 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.