Power law Pólya’s urn and fractional Brownian motion

Abstract

We introduce a natural family of random walks \(S_n\) on \(\mathbb{Z }\) that scale to fractional Brownian motion. The increments \(X_n := S_n - S_{n-1} \in \{\pm 1\}\) 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 \(\mu \) lies in the domain of attraction of an \(\alpha \)-stable subordinator, for \(0<\alpha <1/2\)) the walks scale 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” analogues of simple random walk on \(\mathbb{Z }\).