Lévy Distribution

$${\mathcal F}[P_N(k)]= {\mathcal F}[\mathop{\rm exp}\nolimits (-N\vert k\vert^\beta)],$$

where ${\mathcal F}$ is the Fourier Transform of the probability $P_N(k)$ for $N$-step addition of random variables. Lévy showed that $\beta\in(0,2)$ for $P(x)$ to be Nonnegative. The Lévy distribution has infinite variance and sometimes infinite mean. The case $\beta=1$ gives a Cauchy Distribution, while $\beta=2$ gives a Gaussian Distribution.

See also Cauchy Distribution, Gaussian Distribution