Brownian Motion - Stationary Distributions

Brownian motion Y(t) is stationary: the differences
Y(t + h) - Y(t)
are independent of t.
To illustrate this visually, we sample a Brownian motion simulation and compute increments
incr1 = Y(t1 + h) - Y(t1), incr2 = Y(t2 + h) - Y(t2), ..., incr1000 = Y(t1000 + h) - Y(t1000).
We must take the ti so ti+1 ≥ ti + h; if the sampling increments overlap, we should not expect independence.
Then we plot the points
(incr2, incr1), ..., (incr1000, incr999).
If the increments are independent of one another, the points should lie in an approximately circular cloud, denser near the center.
The left picture shows such a plot for h = 1, the right for h = 2.

Return to Mathematical Properties of Brownian Motion.