From this 2000 point graph of Brownian motion | |||
on the left is the graph of the increments Y(t + 10) - Y(t). We say this is an increment with delay = 10. Compare this with the right graph, the increments Y(t + 50) - Y(t), plotted to the same scale. | |||
To compare the distributions of these increments, divide the range of the increments into 100 bins and plot the histograms of the number of increments in each bin. | |||
On the left is the histogram of the delay = 10 increments, right is the histogram of the delay = 50 increments. | |||
Both are approximately normal. | |||
The second is wider than the first, in fact, its standard deviation is about 2.25 times that of the first. | |||
Note 2.25 is close to √5. |
Return to Mathematical Properties of Brownian Motion.