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 $\displaystyle X_i=\{-1, 1\}, i=1,...,k. $

$\displaystyle \Omega = \prod_{i=1}^k X_i$

$\displaystyle \mu = \prod_{i=1}^k \mu_i$

with prob meas. $\displaystyle \mu_i(-1)=\mu_i(1)=1/2$

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

Now we want to examine the convergence(k-> infinity) of

$\displaystyle \int_{\Omega}f(x)^2\cdot log |(f(x|) d\mu(x)$.

For this purpose we set $\displaystyle y= (x_1 + ...+ x_k)\cdot k^{-0.5}$ and consider functions of the form

$\displaystyle f(x_1,...,x_k) = \phi(y) \text{ with } \phi \in C_c^{\infty}(\mathbb{R})$.

According to the paper, it should now be obvious that the central limit theorem implies the convergence(for k -> $\displaystyle \infty$) of this integral to the integral

$\displaystyle \int_{-\infty}^{\infty} |\phi(t)|^2 log |\phi(t)| dv(t),\text{ where } v(t) = (2\pi)^{-0.5} e^{(-|x|^2)\cdot \frac{1}{2}} dx$ 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