Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent identically distributed random variables. Set $S_{n}=X_{1}+\cdots+X_{n}$.

(i) State the strong law of large numbers in terms of the random variables $X_{n}$.

(ii) Assume now that the $X_{n}$ are non-negative and that their expectation is infinite. Let $R \in(0, \infty)$. What does the strong law of large numbers say about the limiting behaviour of $S_{n}^{R} / n$, where $S_{n}^{R}=\left(X_{1} \wedge R\right)+\cdots+\left(X_{n} \wedge R\right)$ ?

Deduce that $S_{n} / n \rightarrow \infty$ almost surely.

Show that

$$\sum_{n=0}^{\infty} \mathbb{P}\left(X_{n} \geqslant n\right)=\infty$$

Show that $X_{n} \geqslant R n$ infinitely often almost surely.

(iii) Now drop the assumption that the $X_{n}$ are non-negative but continue to assume that $\mathbb{E}\left(\left|X_{1}\right|\right)=\infty$. Show that, almost surely,

$$\limsup _{n \rightarrow \infty}\left|S_{n}\right| / n=\infty$$