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I'm trying to understand what are "Tail Events" from my class notes and in the process I ran into weird examples. They define there a sequence of iid random variables $X_n$ which get the values $\{-1,0,1\}$ with the same probability. The also define $S_n = \sum_{k=1}^{n} X_k$ and look at two sets:

$A_1 = \{S_n = 1 \mbox{ infinitely often} \}$

$A_2 = \{S_n \mbox{ is odd infinitely often} \}$

In my understanding both events can't be tail events because changing $X_1$ will change the sum and its parity, but the notes says that $A_1$ is a tail event while $A_2$ isn't! Can someone explain to me why is that the case?

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    Maybe if you include the reasoning in the notes, we can figure out what the problem is, cause from where I sit, you're right. – 2017-01-20
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    The problem is there is no reasoning in my notes :( – 2017-01-20
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    Okay, your instructor was probably just confused then. Or maybe they meant to do $X_n=1$ infinitely often for the first one, or something. – 2017-01-20

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I agree with you. Both are not tail events.

As per your intuition, a counterexample to both is when $X_1=1$ and all others zero. This outcome is in both $A_1$ and $A_2$ but the if you change it so that $X_1 = 0$ then that outcome is in neither. Thus neither $A_1$ nor $A_2$ is $\mathcal{T}_2$-measurable for $\mathcal{T}_2 = \sigma(X_2,X_3,\ldots).$