Fractional simple random walk

Fractional Brownian motions are the most natural generalizations of ordinary (one-dimensional) Brownian motions that allow for some amount of long range dependence (a.k.a. “momentum effects”). They are sometimes used in mathematical finance as models for logarithmic asset prices. We describe some natural random simple walks on the integers that have fractional Brownian motion as a scaling limit. In a sense, these walks are the most natural discrete analogs of fractional Brownian motion.