=F_k[exp(-N|k|^beta)](x),](/images/equations/LevyDistribution/NumberedEquation1.svg) |
where 
is the Fourier transform of the probability
for 
-step addition of random variables. Lévy showed that
for 
to be nonnegative. The
Lévy distribution has infinite variance and sometimes infinite mean. The case
gives a Cauchy
distribution, while 
gives a normal distribution.
The Lévy distribution is implemented in the Wolfram Language as LevyDistribution[mu, sigma].
