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, |

_{1}K(t - t_{1})_{2}K(t - t_{2})_{3}K(t - t_{3}) |

The amplititudes a_{i} and times t_{i} 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 ^{-d-1}dadt. |

This is called the Levy distribution. |

Here is the sum of 100 such steps, with |

Return to Levy flights.