Power law Polya’s urn and fractional Brownian motion

Abstract

We introduce a natural family of random walks S[subscript n] on Z that scale to fractional Brownian motion. The increments X[subscript n] := S[subscript n] − S[subscript n]−1 ∈ {±1} have the property that given {X[subscript k] : k < n}, the conditional law of X[subscript n] is that of X[subscript n−k[subscript n]] , where k[subscript n] is sampled independently from a fixed law μ on the positive integers. When μ has a roughly power law decay (precisely, when μ lies in the domain of attraction of an α-stable subordinator, for 0 < α < 1/2) the walks scale to fractional Brownian motion with Hurst parameter α + 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” analogues of simple random walk on Z.