[Nesting in] random walks

It is a consequence of the Central Limit Theorem that the pattern of any random walk with steps of bounded length (see page 977) must have a certain nested or self-similar structure, in the sense that rescaled averages of different numbers of steps will always yield patterns that look qualitatively the same. As emphasized by Benoit Mandelbrot in connection with a variety of systems in nature, the same is also true for random walks whose step lengths follow a power-law distribution, but are unbounded. (Compare page 969.)