I am reading the Durret's Probability Theory and Examples.
In the book, tail $\sigma$ field is defined as
$\mathcal T=\cap_n^\infty\sigma(X_n,X_{n+1},…)$.
Then the book shows some easy example as follows:
Let $S_n=X_1+X_2+…+X_n$
$$\{\lim_{n\to\infty}S_n\ exists\}\in\mathcal T$$
$$\{\limsup_{n\to\infty}S_n>0\} \notin\mathcal T$$
$$\{\limsup_{n\to\infty}\frac{S_n}{c_n}>x\}\in\mathcal T,c_n\to \infty$$
I pondered these examples for 2 hours, but still could not understand. I am so damn, can anyone give some intuitive interpretations or a written out proof? thanks in advance.
Best Answer
For the first example note that for a $n\in\Bbb N$ $$\sum_{i=1}^\infty X_i \text{ exists} \Leftrightarrow \sum_{i=n}^\infty X_i \text{ exists}$$ where the right side is clearly $\sigma (X_n , X_{n+1} , \ldots )$ measurable. Since this holds for every $n$, the left side is $\mathcal T$ measurable.