Let $X_1,X_2,…$ be real valued random variables. Put $\mathfrak S_n=\sigma (X_n)$ and $S_n=X_1+X_2+…+X_n$. Let $\mathfrak T_n=\sigma (X_{n+1},X_{n+2},…)$ and define the tail $\sigma$-algebra $\mathfrak T=\cap _n \mathfrak T_n$.
The book I am reading now says : $\{\limsup S_n>b\}\notin \mathfrak T$
But I don't get why this holds. Intuitively, this event seems to be unaffected by first finite happenings… Could you give me an example or explanation? Thank you in advance.
Edit:
For example, let $X_n =^{dist} U_{(-1,1)}$ for all $n$, assuming that they are not independent and let $b=0$.
Is is still true that $\{\limsup S_n>b\}\notin \mathfrak T$?
Best Answer
(This answers the original version of the question.)
Take $X_n(\omega) = 0$ for all $n \ge 2, \omega \in \Omega$. It's a zero-valued function with respect to $\omega \in \Omega$, so it's a real-valued nonnegative random variable, so that $S_n = X_1$ for all $n \in \Bbb N$. It's clear that $S_n$ depends entirely on $X_1$ and $$\{\limsup S_n > b\} = \{\limsup X_1 > b\} \notin \mathfrak T.$$