I've been trying to come up with a intuitive, practical distinction between convergence in probability and convergence almost surely. Can someone please tell me if the following is correct?
Let $X_n$ be a statistic calculated over and over again from many random samples of size $n$ and $X$ the quantity it converges to as $n \rightarrow \infty$. Let $\epsilon$ be an arbitrary boundary around $X$. Let $A_n$ be the number of $X_n : ~ |X_n - X| > \epsilon$ (in other words, the number of $X_n$ outside a given bound) and let $B_n$ be the number of $X_n : ~ |X_n - X| \le \epsilon$ (the number of $X_n$ inside a given bound).
Then, as $n \rightarrow \infty$...
If $X_n \overset{P}{\rightarrow} X$...
- $\frac{A_n}{B_n}\rightarrow 0$
But if $X_n \overset{a.s.}{\rightarrow} X$...
${\color{red} {A_n \rightarrow k < \infty}}$
$\frac{A_n}{B_n}\rightarrow 0$
Is the above a valid way of interpreting the difference between convergence in probability and almost sure convergence? Thanks.