Simulating Levy flights

Some care is needed to simulate Levy flights. Start with the unit step function K(t)

To build a random process with the desired properties, Levy added together many step functions, each multiplied by a factor to change the height of the step, and with the jumps happening at different times.

That is,

a1K(t - t1) + a2K(t - t2) + a3K(t - t3) + ...

The amplititudes ai and times ti can be described as points in an address plane

Adding the step functions with these times and amplitudes gives this function.

To finish the construction, Levy specified the probability of including steps with times and amplitudes in a particular interval.

The probability of finding amplitude a in the interval [a, a + da] and time t in the interval [t, t + dt], is Ca-d-1dadt.

This is called the Levy distribution.

Here is the sum of 100 such steps, with d = 0.9.

Return to Levy flights.