Integrability of a sequence of iid random variables I'd really appreciate some hints on the first part of the following question:
Let $f_n, n\in \mathbb{N}$ be a sequence of iid random variables over $(\Omega, A,P)$. That is, 
$P(\{f_1 \in A\})=P(\{f_2 \in A\})= ···$ for all $A \in \mathbb{B}$. 
Show that, if there is a $c \in \Re$ such that
$P\Bigg(\Bigg\{\lim_{n \to \infty}\frac{1}{n}\sum\limits_{i=1}^n f_i=c\Bigg\}\Bigg) >0,$ then
$P(\{|f_n|>n$ for infinitely many $n\})=0.$
Then conclude that $f_1$ (and therefore also $f_2,f_3,..$ because of the identical distribution) is integrable.
 A: *

*Borel-Cantelli Lemma


You want to show that
$$P(\limsup A_n) = 0$$
where $A_n = \{|f_n| > n\} = \{\frac{|f_n|}{n} > 1\}$
Perhaps we might try Borel-Cantelli Lemma?
If we can show the following
$$\sum_{n=1}^{\infty} P(A_n) = \sum_{n=1}^{\infty} P(\frac{|f_n|}{n} > 1) < \infty$$
Note that $P(A_n^C) = 1 - P(\frac{|f_n|}{n} \le 1) $
then we would get what we want.



*Kolmogorov 0-1 Law


Denote
$$\overline{f_n} := \frac{1}{n}\sum\limits_{i=1}^n f_i$$
to have:
$$P(\lim \overline{f_n}=c) >0$$
Observe that
$$\{\lim \overline{f_n}=c \} \in \bigcap_{n=1}^{\infty} \sigma(f_n, f_{n+1}, ...)$$
By the Kolmogorov 0-1 Law, 
$$P(\lim \overline{f_n}=c) = 0 \ \text{or} \ 1$$
Since
$$P(\lim \overline{f_n}=c) >0$$
we have that
$$P(\lim [\overline{f_n}]=c) = 1$$
which is equivalent to either:
$$P(\liminf [\overline{f_n}]=c) = 1$$
$$P(\limsup [\overline{f_n}]=c) = 1$$
Try using one of those with one of the inequalities below.



*Important inequalities


Williams - Probability with Martingales



Deduced similarly:
(iii) If $\liminf x_n > z$, then
$ \ \ \ \ \ \ \ (x_n > z)$ eventually (that is, for infinitely many n)
(iv) If $\liminf x_n < z $, then
$ \ \ \ \ \ \ \ (x_n < z)$ infinitely often (that is, for infinitely many n)

