I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set $X_1,…,X_n$?
Cause what I don't understand is, the sigma algebra is defined as $$G_{\infty}:= \bigcap_n \sigma ({X_n,X_{n+1},…}).$$
So the intersection also depends on the random variables $X_1,..,X_n$ due to the sigma algebra $\sigma(X_1,X_2,..)$. Therefore, I just don't see how we can say that the events of these finitely many random variables is not important to an event in the sigma algebra. So, where exactly does this interpretation come from?
Best Answer
When an event, call it $T$, is a tail event, it is in $\cap_{n=1}^{\infty}\sigma(X_{n},X_{n+1},...)$, so it is an element is every one of the $\sigma$-algebras $\sigma(X_{n},.X_{n+1},...)$. The reason we can interpret this as not depending on any finite set $\{X_{1},X_{2},...,X_{n}\}$ is because while the event is in $\sigma(X_{1},X_{2},...)$, it is not necessarily dependent on the first few random variables. In fact, it is not dependent on them. We know that $\sigma(X_{n},X_{n+1},...)\subseteq\sigma(X_{1},X_{2},...)$, so omitting the fist $n-1$ random variables did not change whether the event was in the $\sigma$-algebra. If we think about the $\sigma$-algebras as "current knowledge" about a space, we are saying that we can still determine whether or not $T$ occurs just based on the limited knowledge about $\{X_{n},X_{n+1},...\}$ for any $n$.