Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables? I hope the title in itself is clear, if not allow me to give an example.

In Class my Professor did the following:


  Given a sequence $(X_n)_{n \in \mathbb{N}}$ of non-negative i.i.d. RV $X_n \sim X$ with $E(X)&lt; \infty$ then $\frac{X_n}{n} \to 0 $ almost surely


To prove this, it is easy to show that $\sum P (X_n/n &gt; \epsilon) &lt; \infty$ and thus conclude by Borel-Cantelli Lemma that $$P ( \limsup \lbrace X_n/n &gt; \epsilon \rbrace ) =0 \iff P ( \lim \inf \lbrace X_n /n \leq \epsilon \rbrace )=1 $$
But one indeed looks for the answer that $\exists \Lambda \subset \Omega$ with $P( \Lambda)=1$ and for all realizations $\omega \in \Lambda$ we have that $X_n(\omega)/n \to 0$.

In the question Is it correct to say that ($\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) \supseteq \limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}$? Daniel Fischer gives an answer on how to get an inclusion of the form $\liminf \lbrace ... \rbrace \subset \lbrace \limsup ... \rbrace$.   I do believe that he makes use of the (complementary) statement which says that for $\epsilon &gt;0$ $$ \lbrace \lim \sup | X_n - X| &gt; \epsilon \rbrace \subset \lim \sup \lbrace | X_n -X | \geq \epsilon \rbrace \subset \lbrace \lim \sup |X_n - X | \geq \epsilon \rbrace \tag{*} $$
which gives $$ \lbrace \lim \sup |X_n -X | &lt; \epsilon \rbrace \subset \lim \inf \lbrace | X_n - X | &lt; \epsilon \rbrace \subset \lbrace \lim \sup |X_n -X | \leq \epsilon \rbrace $$



My question(s): 1) Are there more chains of inclusions as in (*) that I should know in order to give statements about the $\limsup$ or $\liminf$ of sequence of Random Variables rather than the $\limsup,  \liminf$ of the sequence of Events?

2) In the above chain of inclusions, does the $\epsilon &gt;0$ actually depend on $k$? The Definition gives $$\limsup \lbrace X_n/n &gt; \epsilon \rbrace = \bigcap_{n=0}^\infty \bigcup_{k=n} A_k \text{ where } A_k:= \lbrace X_k/k &gt; \epsilon \rbrace $$ so intuitively I would say that $\epsilon=\epsilon(k)$ varies as $k$ increases. 

3) Are there any general techniques you'd recommend to make a statement about the limit of a sequence of random variables using Borel-Cantelli Lemmas?
Q1,3

Williams - Probability with Martingales







Deduced similarly:

(iii) If $\liminf x_n &gt; z$, then

$ \ \ \ \ \ \ \ (x_n &gt; z)$ eventually (that is, for infinitely many n)

(iv) If $\liminf x_n &lt; z $, then

$ \ \ \ \ \ \ \ (x_n &lt; z)$ infinitely often (that is, for infinitely many n)



Q2

Please elaborate. $\varepsilon$ is supposed to be an arbitrary given. I think $k$ depends on $\epsilon$. For example, $k=5$ might work for $\epsilon = 0.5$ but $k=5$ might not work for $\epsilon = 0.01$ so $k$ would have to instead be say $10$
Q1,3

Williams - Probability with Martingales







Deduced similarly:

(iii) If $\liminf x_n &gt; z$, then

$ \ \ \ \ \ \ \ (x_n &gt; z)$ eventually (that is, for infinitely many n)

(iv) If $\liminf x_n &lt; z $, then

$ \ \ \ \ \ \ \ (x_n &lt; z)$ infinitely often (that is, for infinitely many n)



Q2

Please elaborate. $\varepsilon$ is supposed to be an arbitrary given. I think $k$ depends on $\epsilon$. For example, $k=5$ might work for $\epsilon = 0.5$ but $k=5$ might not work for $\epsilon = 0.01$ so $k$ would have to instead be say $10$