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Under which conditions converges a sum of i.i.d. random variables

$$ \frac{1}{a_N} \sum\limits_{n=1}^N X_n $$

to a symmetric stable distribution? Two examples of sufficient conditions are finite variance or symmetry of $X_n$. But can we say anything about the symmetry of the limit distribution without making any of these two assumptions? Thanks.

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Can we choose the values of $a_N$ or did you have some particular sequence in mind, e.g. $a_N = N$ or $a_N = \sqrt{N}$ for all $N$? – Dilip Sarwate Dec 6 '11 at 16:36
up vote 1 down vote accepted

Please see this answer of mine that gives the reference to the complete specification of the generalized central limit theorem.

In regards to your specific question. $X_i$ have to be symmetric themselves. Let $F_X(x) \sim \vert x \vert^{-\alpha}$ for large $\vert x \vert$. Then, $X_i$ has be symmetric about the origin for $\alpha \geq 1$. If $ 0< \alpha < 1$, $X_i$ can be symmetric about any point.

If $\alpha \not= 2$, then $a_n \sim n^{\min(2, 1/\alpha)}$. If $\alpha = 2$, then $a_n \sim \sqrt{n \log(n)}$, where $\sim$ mean equal up to a constant.

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