Stable distribution

Summary

A stable distribution is a law on $\R$ whose type is preserved under independent sums. A sum of independent copies has the same law up to a change of scale and location. Each stable law has an index $\alpha \in (0, 2]$; the normal distribution is the case $\alpha = 2$ and the Cauchy distribution is a case with $\alpha = 1$. Stable laws are exactly the possible nondegenerate limits of suitably normalised sums of i.i.d. random variables, which extends the central limit theorem to variables of infinite variance. Every stable law is infinitely divisible.