About Example 1.8.2 in Durrett: Probability Theory and Examples The example is about tail $\sigma$-field. Given i.i.d. r.v. $ X_1, X_2, \dots $ and the partial sum $ S_n = X_1 + \dots + X_n $. The example says that 
$\{ \limsup_{n\rightarrow\infty} S_n > 0 \} \notin \mathcal{T}$ and
$\{ \limsup_{n\rightarrow\infty} S_n/c_n > x \} \in \mathcal{T}$ if $c_n \rightarrow \infty$,
where $\mathcal{T} = \cap_n \mathcal{F}'_n = \cap_n \sigma(X_n,X_{n+1},\dots)$ is the tail $\sigma$-field.
I am always getting in trouble with the $\limsup$ stuff. Can anyone provide some explainations about the above example. Many thanks in advance.
 A: Hint: For every $k$, 
$$\limsup\limits_{n\to\infty}\,S_n/c_n=\limsup\limits_{n\to\infty}\,(S_{k+n}-S_k)/c_{k+n}.
$$
This identity uses the fact that $c_n\to\infty$.
A: Durrett mentions that, intuitively, an event is in the tail sigma algebra when it is not influenced by changing the value of a finite number of the $X_n$. Let me handwave just for a bit. Suppose that in the first one the $\limsup$ is positive and finite. Then you could change the value of $X_1$ to a very large negative number, so as to make the $\limsup$ of the sum negative. However, in the second example, any finite number of additions to the values of the $X$ variables will leave the event unchanged, because the large $c_n$ in the denominator will get rid of them eventually.
This may help you: note that, for any fixed value of $k$, 

$\limsup {X_1+\dots+X_n\over c_n}>x$ if and only if the shifted $\limsup{X_{k+1}+\dots+X_{k+n}\over c_{k+n}}>x$. 

Why is that? Suppose that the first $\limsup$ equals $\alpha$. Then you can find arbitrarily large $n_j>>k$ such that 
$${X_1+\dots+X_{n_j}\over c_{n_j}}\to \alpha$$ 
as $j\to\infty.$ Then it only suffices to write
$${X_{1}+\dots+X_{k}\over c_{k+{n_j}}}+{X_{k+1}+\dots+X_{k+{n_j}}\over c_{k+{n_j}}}\to\alpha$$ as $j\to\infty$ to assert the answer, because the first fraction approaches zero. Again: a finite number of values don't affect the result. 
Now, the terms of the original $\limsup$ are not in $\mathcal F'_m$ for $m>k$, but the terms of the $\limsup$ shifted by $k$ will belong to $\mathcal F'_k$ for any $k$. This should be enough to assert the measurability with respect to $\mathcal T$.
