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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.

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  1. 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.


  1. 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.


  1. Important inequalities

Williams - Probability with Martingales


enter image description here


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)


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