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Thread: convergence of iid

  1. #1
    Junior Member
    May 2008

    convergence of iid

    I guess I can use LLN...

    Show that for any sequence $\displaystyle X_1,\dots,X_n,\dots$ of iid random variables, $\displaystyle \frac{1}{n^2}\sum_{i=1}^n iX_i\to \frac{m}{2} \;\;a.s.$ if and only if $\displaystyle X_1$ is integrable and $\displaystyle \mathbb{E}[X_1]=m$.

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  2. #2
    MHF Contributor matheagle's Avatar
    Feb 2009
    First center the random variables and then use Kolmogorov's Three Series and then Kroncker's lemma.
    You can actually quote the literature.
    I proved this type of result in my dissertation for non-independent stochastically dominated rvs a long time ago.

    I found the exact result (exact is a pun by the way).
    See Corollary 3 on page 128 from Chow and Teicher, which follows from Theorem 3.
    You should avoid the three series theorem, because we don't even want to mention having a second moment here.
    Last edited by matheagle; Jun 7th 2010 at 10:33 PM.
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