# Generalized Central Limit Theorem

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Tail behavior of sums of random variables determine the domain of attraction for a distribution. If variance exists, under the central limit theorem (CLT), distributions lie in the domain of attraction of a normal distribution. For infinite variance models one appeals to the generalized central limit theorem (GCLT) and finds that distributions lie in the domain of attraction of a stable distribution. Stated differently, the GCLT states that a sum of independent random variables from the same distribution, when properly centered and scaled, belongs to the domain of attraction of a stable distribution. Further, the only distributions that arise as limits from suitably scaled and centered sums of random variables are stable distributions.

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Contributed by: Roger J. Brown and Bob Rimmer (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The shift and scale values required in order for the distribution to converge to a standardized scale and location are shown with each rendering of the plot.

When , the mean of the stable distribution does not exist, so a different centering strategy is necessary.

This Demonstration has implications for cases where the conventional assumption of finite variance must be suspended. Tail behavior determines the domain of attraction. Extreme value theory, an extension of heavy-tail mathematics, is based on sums of the maxima of sums of random variables.

Resources to explore this more fully with notebooks allowing values across the entire parameter space may be found at www.mathestate.com.

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