Sum of sequence of random variables infinitely often equals 1

Question

Let $ X_1,X_2,… $ be an infinite sequence of independent identically distributed random variables that get the values ${-1,0,1}$ with probability $1/3$. Set $Sn=∑X_i$. I want to show that the event $S_n = 1 $ i.o. (infintely often) is in the tail-sigma-algebra of $ X_1,X_2,… $ Any ideas?

Answer 1

It is not an event in the tail-sigma-algebra since you can take the event $X_2=X_3=...=0$ and still cannot know if it happens or not.

Answer 2

This is not always true. Consider the case with $P[X_1 = 1] = P[X_1 = 0] = 1/2$ and $X_i \equiv 0$ for all $i \geq 2$. Then $P[S_n = 1~\mathrm{i.o}] = P[X_1 = 1] = 1/2$ and therefore is not a tail event.