Power law Polya's urn and fractional Brownian motion

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We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments Xₙ have the property that given Xₖ: k < n, the conditional law of Xₙ is that of Xₙ₋ₖₙ, where kₙ 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.