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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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Almost-sure convergence = convergence on the complement of a set of measure zero, no? –  Harry Gindi Aug 11 '10 at 14:21
    
@Harry: I never took measure theory. Its on my long to read list –  Casebash Aug 11 '10 at 21:24
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Casebash: if you don't know measure theory, then I suggest you learn it fast! Otherwise you can't really appreciate these kinds of problems; probability needs measure theory for proper understanding. –  Zen Harper Aug 12 '10 at 5:29
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Seconding Zen Harper's comment, the answer to your second question is that various formulations of different types of convergence, and the relationships between them, are discussed in detail in most books on measure theory or measure-theoretic probability. Just to name two of my favorites: Real Analysis by Folland and Probability and Measure by Billingsley. (When looking for alternative formulations in books, don't overlook the exercises!) –  Mark Meckes Aug 12 '10 at 13:26
    
I would suggest Rudin's Real 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
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1 Answer

up vote 2 down vote accepted
  1. Maybe I didn't get the notation, but if you suppose, that $\operatorname{dif}_x$ is just $|T_n(x) - T(x)|$, then 1, first and second reformulation, is not correct: suppose all functions on $[0, 1]$, $T \equiv 0$ and $T_n(x) = \chi_{[0, \frac1n]}$ — the characteristic function of $[0, \frac1n]$, then for $d = 1$ and for all $n$ on the set of positive measure ($[0, \frac1n]$) the difference is not smaller then $1$. Third reformulation is correct.

  2. Dyachenko and Ulyanov in their book 'Measure and integration' (unfortunately only in russian or spanish) used similar notions: $F_{k, m} = \left[x \in X: |T_k(x) - T(x)| > \frac1m \right]$: then if we denote $E = \left[x \in X : T_k(x) \to T(x) \text{as} k \to \infty \right]$ we will have $$X \setminus E = \cup_{m=1}^\infty \cap_{n=1}^\infty \cup_{k=n}^\infty F_{k, m}$$

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Do you mean this characteristic function? –  Casebash Aug 12 '10 at 12:44
    
Also, what do you mean by T_n(x)? Do you mean X_n –  Casebash Aug 12 '10 at 12:49
    
1) Not this one , but the first one in the article "Characteristic function", it's more datails there: en.wikipedia.org/wiki/Indicator_function 2) By T_n I mean the characteristic function of [0, 1/n]. –  Kirill Shmakov Aug 12 '10 at 13:58
    
Unfortunately some of my edits may have made your answer not make sense (I reordered statements and changed a d to a e –  Casebash Aug 13 '10 at 13:05
    
@Casebash: I'm not sure how it works over at math.SE, but here, if someone offers a correct answer to the original version of your question, you are strongly encouraged to accept it and re-ask your question. It's not nice to change the goalposts... –  Harry Gindi Aug 13 '10 at 13:10
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