I've found this exercise on the web, which is an application of Hewitt – Savage $0\text{-}1$ law, but I really can't even start it.
Let $(X_i)_i$ i.i.d r.v- and consider the random walk $S_n= X_1 + \cdots + X_n$ on $\mathbb{R}$.
Then one of the following occurs with probability $1$:
- $S_n = 0$ for all $n$
- $S_n \rightarrow \infty$
- $S_n \rightarrow -\infty$
- $- \infty = \liminf S_n \leq \limsup S_n = \infty$
I know that $\{ S_n =0 \text{ i.o.}\}$ is invariant under finite permutations, so $P(\{ S_n =0 \text{ i.o.}\}) \in \{ 0,1\}$.
I also know that $\{ \sum_{k=1}^\infty X_k < \infty\}$ is in the exchangeable $\sigma$-algebra. Hence $\{ \sum_{k=1}^{\infty} X_k = \infty\}$ is also in the same sigma algebra and it occurs with probability either $0$ or $1$.
But then I don't know how to move on.
Best Answer
From https://ocw.mit.edu/courses/mathematics/18-175-theory-of-probability-spring-2014/lecture-slides/MIT18_175S14_Lecture23.pdf
If $X_1$ is a.s. constant, then one of the first 3 is almost sure.
Else, by Hewitt-Savage, for any $t \in [-\infty, \infty]$, then $\limsup S_n \ge t$ has probability 0 or 1. As $t$ goes down, the probability crosses over from 0 to 1 at some point, so there exists some $C \in [-\infty, \infty]$, $\limsup S_n = C$ almost surely. Similar for $\liminf S_n = D$.
Since $$C = \limsup S_n = X_1 + \limsup \sum_{i= 2}^n X_i =X_1 + C \text{ almost surely}$$ and $X_1$ is not a.s. constant, $C$ is not finite. So $C \in \{-\infty, \infty\}$. Similarly for $D$.
So one of the last 3 almost surely happens.