A Generalized Central Limit Theorem
Another important property of stable distributions is the role that they play in a generalized central limit theorem. The central limit theorem states that the sum of a number of independent and identically distributed (i.i.d.) random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution as the number of variables grows. (Voit 2003, § 5.4.3)
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