I was having difficulty in understanding the difference between convergence in probability and almost sure convergence, so I decided to try to reduce them to some sort of canonical form.

- Convergence in probability: ${\lim}_{n \to \infty } Pr(|X_n-X| \ge e)=0$
- Almost sure convergence: $Pr({\lim}_{n \to \infty} X_n=X)=1$

After playing around with the figures, I got the following results.

- Convergence in probability: $\forall e, d, n>N(e,d): dif_x \ge e \text{ with } p < d $
- Almost sure convergence: ($\forall e, n>N(e): dif_x < e) \text{ with } p=1$
- Alternative form: ($\forall e, n>N(e): dif_x \ge e) \text{ with } p=0$

A few notes:

- Here $dif_x$ means how far about points at this location are from the limit
- $N(e,d)$ simply says that we can find a suitable value of N so that this holds which depends on e and d
- The differences between the two types seem more obvious in this form

So, my questions are:

- Is this correct?
- Have reductions into this kind of form been studied? If so, where can I learn more about this?

Real Analysisby Folland andProbability and Measureby Billingsley. (When looking for alternative formulations in books, don't overlook the exercises!) – Mark Meckes Aug 12 '10 at 13:26Real and Complex Analysis. Once you know some ordinary measure theory, translating it to probability-speak is trivial. Almost-sure=Almost-everwhere, Expectation=Integral, etc. – Harry Gindi Aug 13 '10 at 13:08