I have problems to find out how to do discrete simulation of the Lévy walk. I can sum my doubts in a few questions:


1. According to [Wikipedia](https://en.wikipedia.org/wiki/L%C3%A9vy_flight) it seems to me that Lévy flight can be produced just by integration of samples with Lévy distribution, in other words: $x(k+1) = x(k) + v(k)$, were $v(k)$ is a random sample from Lévy distribution. Is this correct?
2. How can I generate a Lévy distribution? Can I convert random samples from other distribution to random samples with Lévy distribution? If yes, please add a reference.
3. What is the difference between Lévy walk and Lévy flight?



Simulation of Lévy walk
1. Indeed, the Lévy flight is a random walk where the step increments $\nu$ are i.i.d. with a Lévy distribution, $p(\nu)\rightarrow 1/\nu^{1+\alpha}$ for $\nu\rightarrow\infty$, with exponent $0<\alpha<2$.
2. [Generating Lévy random variables from Gaussian.](http://andreweckford.blogspot.nl/2011/05/generating-levy-random-variables-from.html)
3. The difference between a Lévy flight and a Lévy walk is whether each step takes the same time irrespective of its length (Lévy flight), or whether the time is proportional to the step length (Lévy walk). A Lévy flight implies an unbounded velocity, which is unphysical. Further reading: [Lévy walks versus Lévy flights.](http://link.springer.com/chapter/10.1007%2F978-94-009-5165-5_29#page-1)


