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Math Help - sLLN vs. wLLN

  1. #1
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    sLLN vs. wLLN

    Hi,

    I would love to have an example for strong vs. weak law of large numbers:

    * the strong law requires iid distribution and  EX_i = \mu such that  \overline {X_n} \to \mu as such.
    * the weak law requires  EX_i \to \mu and  VarX_i \to 0 but does not have to be iid.

    What would be an example of a non-iid function that fails the strong law but holds up with the weak law?

    Thanks!
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  2. #2
    MHF Contributor matheagle's Avatar
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    DO you really want examples of these?
    I have a slew of them.
    See the st petersburg game.

    Your conditions are incorrect.
    The point is, not iid, but whether the mean exists.

    Let your sequence be iid with density

    f_X(x)=x^{-2}I(x>1).

    You can obtain a WLLN but not a SLLN, due to the Borel-Cantelli lemma.
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  3. #3
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    Examples

    Matheagle,

    Thanks for
    1) correcting my conditions, would have missed that;
    2) reference to St. Petersburg game - not sure though where to find it (are you talking about the board game? I checked the arcade and didn't see a St. Petersburg game in there) - can you send me in the right direction? I really would like examples.
    3) saving me a bit of anxiety on my last exam: your suggestion (in response to one of my posts a little while ago) to use mgf's for getting moments stuck in my head and turned out to be really, really useful. ;-)
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  4. #4
    MHF Contributor matheagle's Avatar
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    google st petersburg paradox
    It's 300th anniversary is coming up

    The point to that game was trying to find a fair price when the rvs were
    a transformed geometric...

    P(X=2^n)=2^{-n} n=1,2,3...

    Here the first moment is infinite but barely.
    These are the type of rvs I study.

    Feller Volume 1 page 251-253 has a weak law solution for the st pete game
    BUT that solution is lame.
    Klass and Teicher proved in 78 that the almost sure behaviour is such that...

    \liminf_{n\to\infty}{\sum_{k=1}^nX_k\over {\rm n log_2 n}}=1 almost surely

    That's base 2, by the way.

    and \limsup_{n\to\infty}{\sum_{k=1}^nX_k\over {\rm n log_2 n}}=\infty almost surely


    The Weak Law proved in Feller is that...

    {\sum_{k=1}^nX_k\over {\rm n log_2 n}}\buildrel P\over\to 1

    You can see a Stong Law fails via Borel-Cantelli.


    An easier case is what I mentioned last week.

    f(x)=x^{-2}I(x>1)

    HOWEVER, if you observe a weighted verson of these sums....
    http://www.sciencedirect.com/science...7c34348333de7a
    you can obtain a strong law
    Last edited by matheagle; March 8th 2010 at 11:43 PM.
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  5. #5
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Statistik View Post
    Matheagle,

    Thanks for

    3) saving me a bit of anxiety on my last exam: your suggestion (in response to one of my posts a little while ago) to use mgf's for getting moments stuck in my head and turned out to be really, really useful. ;-)
    What's weird about that comment, was that I hesitated in making it.
    The post was about using the density and not the MGF to derive the fourth moment. In the past I've received criticism for answering a question with a different technique so I thought about it several times before writing my response.

    In any case, I've probably stayed past my expectations (pun included).
    I never had planned to stay this long.
    It's been a year and I should be doing more of my own research anyhow.
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  6. #6
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    mgf's

    Hey,
    I'll appreciate feedback / questions / suggestions of any sort anytime. It's about learning (right? ;-) ), so getting to think outside of the outlined box is good and probably more realistic in the long run anyway.
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  7. #7
    BMW
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    Re: sLLN vs. wLLN

    Quote Originally Posted by matheagle View Post
    Let your sequence be iid with density

    f_X(x)=x^{-2}I(x>1).

    You can obtain a WLLN but not a SLLN, due to the Borel-Cantelli lemma.
    matheagle, could you please tell me where I can find the math you mentioned? If you kindly write the solution briefly, that will be in fact great!
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