Friday, 24 January 2014

limsup of a sequence of sets (or events)

When seen for the first time it can be difficult to immediately grasp the meaning of  $\bigcap_{i=1}^\infty \bigcup_{k=i}^\infty A_k$.

I found that when I was a student and encountering this for the first time. This is one reason I included #7 on Examples Sheet 1. My advice is to pick it apart slowly.

$A_k$ is the event that the $k$th toss is a head.

First: The event $\bigcup_{k=i}^\infty A_k$ is true (occurs) if and only if there is at least one head somewhere amongst the tosses $i,i+1,i+2,\dots$ .

Second: When we put $\bigcap_{i=1}^\infty$ in front we create an event that is true if and only if the above is true for all $i$. So how can the result be true? Well, if and only if there are an infinite number of heads. (There might also be an infinite number of tails.)

In general, $\bigcap_{i=1}^\infty \bigcup_{k=i}^\infty E_k$ is occurs (or is true) if and only if an infinite number of the events $E_1,E_2,\dotsc$ occur.

The idea is similar to the "$\limsup$" of a sequence $\{a_1,a_2,\dots\}$, which is defined as

$\lim_{i\to\infty} \sup_{k\geq i} a_k$.

How about $\bigcup_{i=1}^\infty \bigcap_{k=i}^\infty A_k$? That is the event that eventually we see only heads. Now perhaps you can see that

$\bigcup_{i=1}^\infty \bigcap_{k=i}^\infty E_k \subseteq\bigcap_{i=1}^\infty \bigcup_{k=i}^\infty E_k $.

Here is an article on it that might also help.