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 μon the positive integers. When μhas a roughly power law decay (precisely, when it lies in the domain of attraction of an αstable subordinator, for 0 < α< 1/2) the walk scales 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" analogs of simple random walk on Z.

32 pages. An appendix offering some tentative asset-price interpretations has been added; the submitted version (which does not contain the appendix) will appear in Probab. Theory and Related Fields