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 $$\operatorname{sinc}\, t = \begin{cases} \frac{\sin t}{t} & \text{if } t \neq 0 \\ 1 & \text{otherwise} \end{cases}$$ Determine the pointwise limit of the sequence of functions $(\Phi_{X_n})_{n \geqslant 1}$.
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
$$\operatorname{sinc}\, t = \begin{cases} \frac{\sin t}{t} & \text{if } t \neq 0 \\ 1 & \text{otherwise} \end{cases}$$
Determine the pointwise limit of the sequence of functions $(\Phi_{X_n})_{n \geqslant 1}$.