Let $n$ be a non-zero natural number and $t$ be a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_X(t) = \mathbb{E}(\mathrm{e}^{\mathrm{i}tX})$.
Show $$\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right).$$

Let $n$ be a non-zero natural number and $t$ be a real number. Using the result of Q1, deduce $$\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}.$$

Let $n$ be a non-zero natural number and $t$ be a real number. We have $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$ and $\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}$.
Determine the pointwise limit of the sequence of functions $\left(\Phi_{X_n}\right)_{n \geqslant 1}$.

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_X(t) = \mathbb{E}(\mathrm{e}^{\mathrm{i}tX})$.
Show $$\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right).$$

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$.
Deduce $$\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}.$$