TITLE:  limsup and liminf for a sequence of sets
QUESTION [8 upvotes]: how does limsup and liminf for a sequence of sets, apply to probability theory. any real world examples would be much appreciated

REPLY [3 votes]: Here's another simple example, in a similar vein as has2's above. Let $X_n$ be a sequence of independent, identically-distributed exponential variables, i.e., $$\mathbb P(X_n > u) = e^{-\lambda u},$$ for some real $\lambda > 0$.  Let $E_n$ be the event that $X_n > n$, and let $E = \limsup E_n$, that is, the event $E$ occurs if there's an infinite (random) subsequence $n_k$ such that $X_{n_k} > n_k$.
We compute $$\sum \mathbb P(E_n) = \sum e^{-\lambda n} < \infty,$$ thus by the Borel-Cantelli lemma, $E$ has probability zero.  With probability one, there exists a (random) number $N$ such that for all $n \ge N$, $X_n \le n$.
Let's analyze this graphically.  Make a plot with the horizontal axis representing time $n$, and the vertical axis $x = X_n$.  Draw the line $x = n$.  For small times $n$, these random points might jump above the line $x = n$.  But the argument above shows that there is some (random) time $N$ after which the points $X_n$ all lie below the line $x = n$.