Another simple example, but here we go. For this project, the "walker" moves around the screen based on a Perlin Noise function that gives "smooth" random numbers so that motion looks more natural. Here are a few versions that I created. Some simply map the noise to a point somewhere on the screen while others use the noise to decide which direction the point will move. This second method allows me to choose my starting point and also have the point more easily reach the edge of the window. I imagine a snail in a box, crawling around and getting stuck in the corners.
Some of the other examples play with the alpha of the point or the background giving the feeling that the point is being drawn with a pencil or that the point is a shooting start with a tail.