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 $S_{n} = X_{1} + \cdots + X_{n}$, $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$, and $$\mathcal{Z}_{n} = \left\{\omega \in \Omega, \exists k \geqslant n, U_{k}(\omega) \geqslant \frac{1}{\sqrt{k}}\right\}$$ By considering $Z = \bigcap_{n \in \mathbb{N}^{*}} \mathcal{Z}_{n}$, show that $\left(\frac{S_{n}}{n}\right)$ converges almost surely to $0$.
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 $S_{n} = X_{1} + \cdots + X_{n}$, $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$, and
$$\mathcal{Z}_{n} = \left\{\omega \in \Omega, \exists k \geqslant n, U_{k}(\omega) \geqslant \frac{1}{\sqrt{k}}\right\}$$
By considering $Z = \bigcap_{n \in \mathbb{N}^{*}} \mathcal{Z}_{n}$, show that $\left(\frac{S_{n}}{n}\right)$ converges almost surely to $0$.