The Classification of Random Walk

Abstract

The simplest definition of random walk is an analytical one. It has nothing to do with probability theory, except insofar as probabilistic ideas motivate the definition. In other words, probability theory will “lurk in the background” from the very beginning. Nevertheless there is a certain challenge in seeing how far one can go without introducing the formal (and formidable) apparatus of measure theory which constitutes the mathematical language of probability theory. Thus we shall introduce measure theory (in section 3) only when confronted by problems sufficiently complicated that they would sound contrived if expressed as purely analytic problems, i.e., as problems concerning the transition function which we are about to define.