Power law Polya’s urn and fractional Brownian motion
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Sheffield_Power law.pdf
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Author(s)
Hammond, Alan
Sheffield, Scott Roger
Date Issued
December 2012
Journal
Probability Theory and Related Fields
Publisher
Springer-Verlag
Citation
Hammond, Alan, and Scott Sheffield. “Power Law Polya’s Urn and Fractional Brownian Motion.” Probability Theory and Related Fields 157, no. 3–4 (December 11, 2012): 691–719.
Version
Original manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00440-012-0468-6
