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Math Help - problem about almost sure and L1 convergence

  1. #1
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    problem about almost sure and L1 convergence

    Can anyone give me some advice about this problem? Thanks.

    Let \lim_{n \to \infty}X_{n}=X a.s.

    And let Y=\sup_n|X_{n}-X|.

    • Proove that Y<\infty a.s.
    • Let Q a new probability measure defined as it follows:
      \displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right], where \displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right].
      Proove that X_{n} \rightarrow X (in L_{1}(Q)).
    Last edited by yavanna; November 11th 2009 at 10:43 AM.
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  2. #2
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    Quote Originally Posted by yavanna View Post
    Can anyone give me some advice about this problem? Thanks.

    Let \lim_{n \to \infty}X_{n}=X a.s.

    And let Y=\sup_n|X_{n}-X|.

    • Proove that Y<\infty.
    • Let Q a new probability measure defined as it follows:
      \displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right], where \displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right].
      Proove that X_{n} \rightarrow X (in L_{1}).
    It should be pointed out that the first question is just a question about real sequences (i.e., not probability) : prove that a convergent sequence is bounded, and explain why it is sufficient to conclude.

    As for the second question, be very careful! It is not L^1, but L^1(Q) that you probably mean; this is very different. The only advice then is to use the bounded convergence theorem; that should do the trick pretty neatly.
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  3. #3
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    Thanks!

    Quote Originally Posted by Laurent View Post
    It should be pointed out that the first question is just a question about real sequences (i.e., not probability) : prove that a convergent sequence is bounded, and explain why it is sufficient to conclude.

    As for the second question, be very careful! It is not L^1, but L^1(Q) that you probably mean; this is very different. The only advice then is to use the bounded convergence theorem; that should do the trick pretty neatly.
    Yes, you're right... It was L_{1}(Q).


    But the convergence to be verified in the first question is a.s.

    Quote Originally Posted by yavanna View Post

    • Proove that Y<\infty a.s.
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  4. #4
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    Quote Originally Posted by yavanna View Post
    But the convergence to be verified in the first question is a.s.
    Yes. The assumption means that for \omega in an event of probability 1, the sequence (X_n(\omega))_n of real numbers converges. However, this convergence implies that the sequence is bounded. Thus, you have the inclusion of events : \{\lim_n X_n= X\}\subset\{Y<\infty\}. If the first one has probability 1, then the same holds for the second one.
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