Is it known what the next-to-leading order term is in the variance of the central limit distribution for the average of $N$ variables each of which is distributed according to $P(x) \sim 1/x^{1+\alpha}$ for $1 < \alpha$?


So we have $\bar{x} = N^{-1} \sum\_{i=1}^N x\_i$ where $x\_i$ are independent and distributed according to $P(x)$, then the distribution of $\bar{x}$, $P\_N(\bar{x})$, will have a variance $\sigma\_N$ which goes to zero as $N\to \infty$ and the first terms are


$\sigma\_N \sim 1 / N^{1-1/\alpha}$ for $1 < \alpha < 2$


$\sigma\_N \sim ( \log(N) / N )^{1/2}$ for $\alpha = 2$


$\sigma\_N \sim 1 / N^{1/2}$ for $2 < \alpha$


What I couldn't find so far is the form of the next-to-leading order terms in particular for $1 < \alpha \leq 2$ (for $2 < \alpha$ I actually found it). I suspect that for $\alpha = 2$ we will have


$$\sigma\_N \sim ( \log(N) / N )^{1/2} ( 1 + c / \log(N) + \ldots)$$


but I haven't found this in any of the probability text books so far.



Variance of central limit distribution for $P(x) \sim 1/x^{1+\alpha}$ for finite but large $N$?
You are looking for a generalization of the [Edgeworth expansion](https://en.wikipedia.org/wiki/Edgeworth_series) to distributions that have slowly decaying tails. 


For $\alpha>2$ this can be found here:


[Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities](http://arxiv.org/abs/1103.4306), by Henry Lam, Jose Blanchet, Damian Burch, Martin Z. Bazant (2011).


It seems that Vinogradov also considers the case $\alpha=2$ in his 1994 book on [Refined Large Deviation Limit Theorems](http://www.amazon.ca/Refined-Large-Deviation-Limit-Theorems/dp/058225499X), but I have only seen the table of contents.


No pointers to $\alpha<2$, regrettably.


