Let $n$ be a non-zero natural 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 $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}(\cos(t X_n)) \end{aligned}$$ Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ converge uniformly on $\mathbb{R}$?
Let $n$ be a non-zero natural 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
$$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}(\cos(t X_n)) \end{aligned}$$
Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ converge uniformly on $\mathbb{R}$?