A full proof of Berry-Esseen inequality in the Central Limit Theorem

[giscus[bot]]

A full proof of Berry-Esseen inequality in the Central Limit Theorem

This theorem gives us the maximum convergence limit of the basin of attraction in the central limit theorem.

https://www.physicslog.com/blog/2018/04/proof-of-berry-essen-theorem/

[physicslog]

Originally posted by John on 2022-05-29 12:49:11

I don't feel your approach. If $F_n$ is CDF (as it states in theorem) then characteristic function $\psi(t)$ is definitely not integral $\int_{-\infty}^ {\infty} F_{n}(x) e^{itx}dx$ on reals...

[physicslog]

Originally posted on 2022-06-03 12:38:59

Hi John, $\psi(t)$ is defined from the definition of characteristic function which is defined as an expected value of $e^{itx}$. I hope a quick Wikipedia article will help.

[physicslog]

Originally posted by Warren Smith on 2023-01-10 00:39:47

Re $m< \frac{2}{5}$, in fact best possible $m=\frac{1}{\sqrt{2\pi}} = 0.3989422804014327$ and Feller chose $\frac{2}{5}=0.4$ since made calculation simpler and it is pretty close to best possible. Hence your final constant $2.668$, if valid, could be shrunk to $2.663$. Allegedly the much better constant 0.469 is valid but the proof is much harder than Feller's. At the time Feller wrote it, his claimed constant 3 was pretty good.

[physicslog]

Originally posted on 2023-01-10 02:37:02

Hi Warren, thanks for your comment. Can you share us the reference where did you find $m = \frac{1}{\sqrt{2\pi}}$?

[physicslog]

Originally posted by Warren Smith on 2023-01-10 16:54:59

It is just because the standard normal density peak value is $\frac{1}{\sqrt{2\pi}}$. This is the max slope of the CDF. $\text{density}(x)= \frac{\exp(\frac{-x^2}{2})}{\sqrt{2\pi}}$.

[physicslog]

Originally posted on 2023-01-11 01:23:52

Thanks for the explanation. Indeed, it makes sense.

[wyt0125]

Hi,teacher,so why is this?∣ψ(t)−1+t^2sigma^2/2|<=| ψ(t)|+|t^2sigma^2/2| called Cauchy Schwarz Inequality in your text? Where is the '1'? Why not the ∣ψ(t)−1+t^2sigma^2/2|<=| ψ(t)|+|t^2sigma^2/2|+1?,Thanks