Central Limit Theorem Application

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May 2nd 2012, 09:50 AM
slabic

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 gauß-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
May 2nd 2012, 11:11 AM
girdav

Re: Central Limit Theorem Application

What do we know about $f$ ?
May 2nd 2012, 11:36 AM
slabic

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.
May 4th 2012, 07:30 AM
slabic

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 gaußmeasure. 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 $\int_{-\infty}^{\infty} |\phi(t)|^2 log |\phi(t)| dv(t)$ .