We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$. We consider the function $\varphi_{n}$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\varphi_{n}(t) = E\left(\cos\left(S_{n} t\right)\right)$ for all real $t$. Show that $\varphi_{n}(t) = (\cos t)^{n}$ for all integers $n \in \mathbb{N}^{*}$ and all real $t$.
We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$,
$$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$
For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$.
We consider the function $\varphi_{n}$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\varphi_{n}(t) = E\left(\cos\left(S_{n} t\right)\right)$ for all real $t$.
Show that $\varphi_{n}(t) = (\cos t)^{n}$ for all integers $n \in \mathbb{N}^{*}$ and all real $t$.