To paraphrase a great quote from "The Shawshank Redemption": Get busy learning or get busy dying.
At the beginning of the summer, I decided to relearn Probability Theory from the very basic. In the past, I always struggled with the axioms of probability theory, especially concepts such as σ-algebra. Now with the quiet summer in Fribourg, I finally have time to fill in these gaps in my knowledge space.
Of course, as usual, it turns out I know much less than I don't know. After gulp down many unfamiliar concepts from measure theory and abstract algebra, I finally arrived at the part where I could write some code to help me improve my intuition (without actually doing the proof).
The following is a numerical demonstration of the three Arcsine Laws for random walks (If there is any problem displaying, you can find the original code on Gist).
I am still working through Rick Durrett's "Probability: Theory and Examples". It is indeed a great but very difficult book - no wonder the reviews on Amazon are a bit low. I am also following the lecture notes from Scott Sheffield, which are mostly based on Durrett's and extreamly helpful. I migth post some more related codes as visual proof before the end of the summer.