Wednesday, August 12, 2009
A Thousand Words
A picture might help help clarify the idea about random walks in multidimensional spaces. Here A is some state of affairs in the past, and B is the present state of affairs. There is a region around B that represents the possible states of affairs some relatively short time in the future. In two (or more) dimensions, most of these next states are further from A than is B. If the direction of the next steps we actually take is purely random, we will most likely end up moving further away from A. I.e. a tendency to move further and further from the way things were in the past does not imply any consistent direction of movement.