I have the following: Let $(X_n)$ be a sequence of i.i.d. random variables.
(a) Assume $\frac{1}{n} S_n=\frac{1}{n} \sum_{i=1}^n X_i$ converges a.s. to a real-valued random variable $Y$. Show that $E[\mid X_1\mid]< \infty$ and that $Y$ is a.s. constant.
(b) If $\frac{1}{n}S_n$ does not converge, show that $limsup_{n \rightarrow \infty} \frac{1}{n}S_n$ and $liminf_{n \rightarrow \infty} \frac{1}{n}S_n$ are still a.s. constant.
To (a): I thought of the following: Let's say $\frac{1}{n}S_n$ converges to a real number, say $c$, then I have:
$\frac{X_n}{n}=\frac{S_n - S_{n-1}}{n}=\frac{S_n}{n}-\frac{n-1}{n}\frac{S_{n-1}}{n-1} \rightarrow c-c=0$, so since $\frac{X_n}{n} \rightarrow 0$ a.s., it follows, that $E[\mid X_1 \mid] < \infty$.
The problem is, that I assumed convergence to a number and not to a random variable. How can I show this and how can I show (b)? Can someone help me?
By Kolmogorov's 0-1 law, any tail measurable random variable ($\lim S_n/n$, $\limsup S_n/n$, and $\liminf S_n/n$ are all tail-measurable) is almost surely deterministic. See page 46 of Williams' Probability with Martingales if you have it. This immediately answers both parts of your question.