Brownian motion Y(t) is stationary: the differences | |||

are independent of t. | |||

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

_{1} = Y(t_{1} + h) - Y(t_{1})_{2} = Y(t_{2} + h) - Y(t_{2})_{1000} = Y(t_{1000} + h) - Y(t_{1000}) | |||

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

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. | |||

The left picture shows such a plot
for | |||

Return to Mathematical Properties of Brownian Motion.