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Math Help - Central Limit Theorem Application

  1. #1
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    Central Limit Theorem Application

    Hello everybody,
    I do not understand an argument in a paper and I hope you could help me.
    It is an application of the central limit theorem and I think it is not that difficult.

    Consider the two-point spaces X_i=\{-1, 1\}, i=1,...,k.
    \Omega = \prod_{i=1}^k X_i
    \mu = \prod_{i=1}^k \mu_i
    with prob meas.  \mu_i(-1)=\mu_i(1)=1/2


    Furthermore let x_i be the coord.fct., i.e. x_i(\pm 1)=\pm 1


    Now we want to examine the convergence(k-> infinity) of
    \int_{\Omega}f(x)^2\cdot log |(f(x|) d\mu(x).
    For this purpose we set y= (x_1 + ...+ x_k)\cdot k^{-0.5} and consider functions of the form
    f(x_1,...,x_k) = \phi(y) \text{ with } \phi \in C_c^{\infty}(\mathbb{R}).


    According to the paper, it should now be obvious that the central limit theorem implies the convergence(for k -> \infty) of this integral to the integral
    \int_{-\infty}^{\infty} |\phi(t)|^2 log |\phi(t)| dv(t),\text{ where } v(t) = (2\pi)^{-0.5} e^{(-|x|^2)\cdot \frac{1}{2}} dx is the gau▀-measure.


    I see roughly the connection between the statement and the CLT, but I am not really able to prove this. I would be pleased, if anybody could help me,

    best regards,
    slabic
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  2. #2
    Super Member girdav's Avatar
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    Re: Central Limit Theorem Application

    What do we know about f?
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  3. #3
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    Re: Central Limit Theorem Application

    In the general context f is in the Sobolev space H^1(R, v), i.e. in L^2(v) and the weak derivative |\nabla f(x)| is in L^2(v), too.
    The idea to prove the general inequation is to prove it for functions with compact support at first-afterwards we use the density for the general case. And in order to do that we consider f(...)= phi(y), as I have written above. And one main argument in this context is the convergence of the integral by using the CLT.
    Last edited by slabic; May 2nd 2012 at 11:51 AM.
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  4. #4
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    Re: Central Limit Theorem Application

    The problem is done.

    One can write \int_{\Omega}f(x)^2\cdot log |(f(x|) d\mu(x) as \int_{-\infty}^{\infty}\phi(x)^2\cdot log (|\phi(x)|) d\mu(y^{-1}(x)), where d\mu(y^{-1}) is the induced measure of the distribution function. Now the CLT implies the weak convergence of d\mu(y^{-1}) to the gau▀measure. Due to the compact support and the fact that one can extend x^2logx continuously to Zero, we have an continuous and bounded integrand and thus the desired convergence to the integral \int_{-\infty}^{\infty} |\phi(t)|^2 log |\phi(t)| dv(t).
    Last edited by slabic; May 4th 2012 at 07:38 AM.
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