Power law Polya's urn and fractional Brownian motion

Abstract

We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments X_n have the property that given {X_k: k < n}, the conditional law of X_n is that of X_{n-k_n}, where k_n is sampled independently from a fixed law \mu on the positive integers. When \mu has a roughly power law decay (precisely, when it lies in the domain of attraction of an \alpha stable subordinator, for 0 < \alpha < 1/2) the walk scales to fractional Brownian motion with Hurst parameter \alpha + 1/2. The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense we describe, they are the natural "fractional" analogs of simple random walk on Z.