fa.functional analysis - Reducing limits to a canonical form

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:

1. Here $dif_x$ means how far about points at this location are from the limit


2. $N(e,d)$ simply says that we can find a suitable value of N so that this holds which depends on e and d


3. The differences between the two types seem more obvious in this form

So, my questions are:

1. Is this correct?


2. Have reductions into this kind of form been studied? If so, where can I learn more about this?

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