Scaling behavior Levy flight (distance from the origin v number of steps)

Question

In the question

• Numerical approximation of Levy Flight

the implementation of a Levy-flight random walk with Matlab was discussed.

For a classical random walk (Brownian motion), we have that the distance from the origin of the walk ($D$) scales as the square root of the number of steps (N): $D \sim N^{1/2}$. What is the analogous result for $\alpha$-Levy flights?