Brownian Motion - Self-Affinity

Here is a more analytical approach.

First, generate a list of, say, 1000 increments Y(t + h) - Y(t).

Divide the range between the minimum and maximum increments into 100 bins and determine how many of the increments fall into each bin.

Then plot the cumulative distribution: the probability of obtaining a value ≤ that represented by each bin.

Here is a picture of the cumulative distribution for h = 1 increments of a 2000 point Brownian motion simulation.

For example, the selected point shows Pr(Y(t+1) - Y(t) ≤ 1.3) = 0.9

To illustrate self-affinity, we compare the cumulative distributions for Y(t + 1) - Y(t) (red dots) and for Y(4⋅(t + 1)) - Y(4⋅t) (blue dots).

For positive u, the blue dots are shifted to the right by a factor of 2 = √4; for negative u the blue dots are shifted to the left by a factor of about 2.

That is, Prob(Y(t + 1) - Y(t) < u) = Prob(Y(4⋅(t + 1)) - Y(4⋅t) < 2⋅u).

Return to Brownian Motion Self-Affinity.