Transition probabilities: Random walk

Abstract

A key link between wavelets and fractals on the analysis side and random walk on the probability side is to be found in the use of filters from signal processing. For the standard dyadic wavelets on the real line, we already sketched this approach in Chapter 1. Stepping back and taking a more general and systematic view of the underlying idea, one sees that in a real sense it is (almost) ubiquitous in both pure and applied mathematics.

The Cat only grinned when it saw Alice.⋯ “Cheshire Puss,” she began, rather timidly, as she did not at all know whether it would like the name: however, it only grinned a little wider. “Come, it’s pleased so far,” thought Alice, and she went on. “Would you tell me, please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the Cat. “I don’t much care where—” said Alice. “Then it doesn’t matter which way you go,” said the Cat. “—so long as I get somewhere,” Alice added as an explanation. “Oh, you’re sure to do that,” said the Cat, “if you only walk long enough.” —Lewis Carroll