Power law Polya's urn and fractional Brownian motion

TL;DR

This paper introduces a family of random walks on integers that scale to fractional Brownian motion, characterized by power law distributed increments, and demonstrates their properties and simulation advantages.

Contribution

It presents a new class of random walks with power law increments that converge to fractional Brownian motion, extending classical models with natural fractional analogs.

Findings

Walks scale to fractional Brownian motion with Hurst parameter lpha+1/2

Increments satisfy an FKG inequality

Walks are easy to simulate and are natural fractional analogs of simple random walk

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.