What do we know about ?
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
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 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
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 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 .