# Beyond the CLT: guarantees on the shape of the sample mean distribution?

If the L.L.N. tell us where our sample mean is going, and the C.L.T. "extend it" telling us how fast the variance is decaying, do we have an other tool telling us how fast the shape (or the further moments) of the distribution of our sample mean is converging ?

While the C.L.T. gives us guarantees about the variance of the distribution of our sample mean for a large but finite n (-or so at least we use it for, creating C.I., estimating errors in hypothesis tests, etc..), we have nothing to estimate our "guarantee" concerning the shape of the distribution. We always assume it is Normal, but that's only the asymptotic shape, we really know nothing about how fast the finite cases are approaching the asymptotic one.

The problem is that the C.L.T. uses information about the original distribution variance, but not about the original distribution shape, and we know that the shape of the sample mean depends on the shape of the original population distribution.. If it is already normal, just with n=1 we already have a Normal sample mean, but if the population distribution has multiple peeks, the C.L.T. has nothing to tell us how fast we are approaching to a Normal, and our intervals may be wrong. We have "guarantees" about the variance, but not about the shape.

So my question, are there other "tools" that can gives us guarantees about the shape - I guess in terms of further moments - of our large but finite n sample mean ?

• see en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem which provides a rate of convergence in terms of the first three moments Oct 31, 2019 at 8:39
• @user257566 I think if you would summarize it, that would make for a good answer. Oct 31, 2019 at 10:51