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  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 $\epsilon &gt;0$  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?
I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous.     Statement: Let $(X_n)_{n \geq 1}$ be a sequence of non-negative random variables that are identically distributed $X_n \sim X \geq 0 $ with $E(X)&lt; \infty$, show that $ X_n/n \to 0$ almost surely     Proof: Since $X_n \sim X$ for some non negative random variables we can easily compute that for $\epsilon &gt;0$ we have $$ \infty&gt;E(X/\epsilon) = \int_0^\infty P(X/ \epsilon &gt; t) dt = \sum_{n=1}^\infty \int_{n-1}^n P(X/ \epsilon &gt; t ) dt \geq \sum_{n=1}^{\infty} P(X/ \epsilon &gt;n) $$ Thanks to the Borel-Cantelli Lemma we obtain that $$ P ( \limsup_{n \to \infty} \lbrace X_n/n &gt; \epsilon \rbrace ) = 0 \iff P ( \liminf_{n \to \infty} \lbrace X_n/n \leq \epsilon \rbrace) = 1 $$     Question: Does the $\epsilon$ in the above depend on $k$?   I think it does and I would argue as follows $$P( \liminf_{n \to \infty} \lbrace X_n/n \leq \epsilon \rbrace )= P  \left( \bigcup_{n=0}^\infty \bigcap_{k=n}  \lbrace X_k/k \leq \epsilon (k) \rbrace \right) $$ Now using the chain of inclusions that state that we have for $\epsilon &gt;0$  $$ \lbrace \limsup |X_n-X| &lt; \epsilon \rbrace \subset \liminf \lbrace |X_n-X| &lt; \epsilon \rbrace \subset \lbrace \limsup |X_n-X| \leq \epsilon \rbrace $$ We conclude that $$ P ( \lbrace \limsup_{n \to \infty } X_n/ n \leq \epsilon(k) \rbrace)=1 $$ Where I choose the same $k \in \mathbb{N}$ as above, and then choose $k^*$ large enough such that $\epsilon(k) \leq \frac{1}{k^*}$, define the set $$ \Lambda_{k^*}:= \lbrace \limsup_{n \to \infty} X_n/n \leq 1/k^* \rbrace \implies P(\Lambda_{k^*})=1$$ and also we obtain that $\Lambda:=\bigcap_{j \geq k^*} \Lambda_{k^*}$ is still an almost sure event, i.e. $P(\Lambda)=1$ but for all realizations $\omega \in \Lambda$ we have $\limsup_{n \to \infty} X_n( \omega)/n \leq 0  \implies X_n( \omega)/n \to 0$ for all $\omega \in \Lambda$ that is$X_n/n \to 0$ almost surely $\Box$  Are my arguments given above correct and more importantly complete? Are there steps where you'd use different arguments?
I have a list of integers, List&lt;Integer&gt; and I'd like to convert all the integer objects into Strings, thus finishing up with a new List&lt;String&gt;.  Naturally, I could create a new List&lt;String&gt; and loop through the list calling String.valueOf() for each integer, but I was wondering if there was a better (read: more automatic) way of doing it? I found an asnwered question concerning this topic already here:  Copying \captionsetup[subfigure]{justification=centering} doesn't help me thought for I am using different packages.  I would like my Captions and Subcaptions to be justified as the text below. Just under my pictures. Any suggestions?  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Find the area ratio between ABC and the hatched triangle.    Thanks in advance. anyone tell how is this effect created which change lips color to orange and little orange on face without brush etc tool how to change the only lips color to orange without brush etc tool and how this is created and little orange on face What is the fastest way to get money in GTA San Andreas? There are quite some houses and clothes for sale, but I do not have any money. What is the fastest way to get money in GTA San Andreas? Can you raise $\pi$ to a real power to make it rational? not working in Phonegap app on Android 2.3.3 How to clone/copy all file/directory attributes onto different file/directory? I want to copy the attributes (ownership, group, ACL, extended attributes, etc.) of one directory to another but not the directory contents itself.  This does not work:  cp -v --attributes-only A B cp: omitting directory `A'              Note: It does not have to be cp. 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