Brownian motion Y(t) has
independent increments: the
differences _{4}) - Y(t_{3})_{2}) - Y(t_{1})_{1} < t_{2} < t_{3} < t_{4}. |

To illustrate this visually, we sample a Brownian motion simulation and compute increments |

_{1} = Y(t_{2}) - Y(t_{1})_{2} = Y(t_{4}) - Y(t_{3})_{1000} = Y(t_{2000}) - Y(t_{1999}) |

We must take the t_{i} so _{i+1} > t_{i} |

Then we plot the points |

_{2}, incr_{1})_{1000}, incr_{999}) |

If the increments are independent of one another, the points should lie in an approximately circular cloud, denser near the center. Here is an example plot. |

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