What do we know about ?
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
with prob meas.
Furthermore let be the coord.fct., i.e.
Now we want to examine the convergence(k-> infinity) of
For this purpose we set and consider functions of the form
According to the paper, it should now be obvious that the central limit theorem implies the convergence(for k -> ) of this integral to the integral
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,
In the general context is in the Sobolev space H^1(R, v), i.e. in L^2(v) and the weak derivative 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.
The problem is done.
One can write as , where is the induced measure of the distribution function. Now the CLT implies the weak convergence of 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 .