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Math Help - almost sure convergence

  1. #1
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    almost sure convergence

    Let X_{1},X_{2}... be iid random variables with mean \mu and E(X_{1}^{4})=1. Set \bar{X}_{n}=(X_{1}+...+X_{n})/n. Show \bar{X}_{n}\to\mu almost surely as n\to \infty.


    So far all I have is convergence in probability from the weak law of large numbers. I don't see what the trick is to using the expected value.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Re: almost sure convergence

    Have you seen the Marcinkiewicz_Zygmund Law of Large Numbers?
    All you need is a first moment.
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  3. #3
    Moo
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    Re: almost sure convergence

    But maybe his exercise is to prove it with the 4th moment !!
    Which is a common proof, but the margin is too narrow to put it down. At least right now
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