I have the following problem:
Let $(X_n)$ be a sequence of rv and $\tau = \bigcap_{n\geq 1} G_n$ Where $G_n= \sigma (X_n,X_{n+1},…)$.
The point of the exercise is to check if a certain event is in the tail $\sigma$-field.
I have the event:
$\{limsup S_k >0\}$ $S_k= \sum_{n=1}^k X_n$
I found the following example:
Tanking $\Omega = [0,1]$ U~unif(0,1) $X_1=U-1/2$, $X_n=0\forall n$
So:$\{limsup S_k >0\}$ =(1/2,1] which is not in the tail sigma field.
I'm not ok with the conclusion of this example since U-1/2 $\in[-1/2,1/2]$..
Does anyone know a formal proof?
I tried:
Define $E_k=\{S_n>0\}$
Then considering following…
$\bigcap_{n\geq 1}\bigcup_{k\geq n} E_k$
I have:
$E_k \in G_k$
$\bigcup_{k\geq n} E_k\in G_n$
And so $\bigcap_{n\geq 1}\bigcup_{k\geq n} E_k \in G_1$
Which is clearly wrong some where..
Thanks
Best Answer
Surely you can check that $$[U\leqslant\tfrac12]=[\forall n,S_n\leqslant0],\qquad[U\gt\tfrac12]=[\forall n,S_n\gt0]=[\exists x\gt0,\forall n,S_n\geqslant x] $$ hence $$ [\limsup S_n\gt0]=[U\gt\tfrac12]. $$ In full generality, the random variable $\limsup S_n$ is not measurable with respect to the tail sigma-algebra of the sequence $(X_n)$.