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Limit of sums of iid random variables which are not square-integrable

Consider a sequence of iid random variables $X_i$ on $\mathbb{R}$ for which $E(|X_i|^\alpha) < \infty$ for $\alpha < 2$, but $E(X_i^2) = \infty$. Let $S_n = \sum_{i=1}^n X_i$. Examples for such distributions include the t-distribution with 2 degrees of freedom with density $f(x) \sim (2+x^2)^{-3/2}$ and inverse gamma distributions with shape parameter 2, $f(x) \sim e^{-1/x} x^{-3}$. The question is what the appropriate Central Limit Theorem is. More precisely, under what conditions can we infer that there are sequences of real numbers $a_n, \, b_n$ and a random variable $\Xi$ such that weakly (in distribution) $ a_nS_n + b_n \to \Xi \, ? $ Note that the usual central limit theorem does not apply, since the variance is infinite. On the other hand, $\Xi$ should not be any particular $\alpha$-stable distribution with $\alpha < 2$.

Do we have to weaken the notion of convergence or do we have to generalize the normalization procedure, or does the CLT simply not hold in this situation?

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    Thank you, this answers it. And I should have known this :) I suspect that if the density $f$ is minimally well-behaved (regular varying?) at $\pm \infty$, then $a_n = (n F(n)}^{-1/2}$ should work where $F(t) = \int^t s^2f(s) ds$.2011-12-11

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