Power law Plya's urn and fractional Brownian motion
成果类型：
Article
署名作者：
Hammond, Alan; Sheffield, Scott
署名单位：
University of Oxford; Massachusetts Institute of Technology (MIT)
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-012-0468-6
发表日期：
2013
页码：
691-719
关键词：
摘要：
We introduce a natural family of random walks S-n on Z that scale to fractional Brownian motion. The increments X-n := S-n - Sn-1 is an element of {+/- 1} have the property that given {X-k : k < n}, the conditional law of X-n is that of Xn-kn, 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 Z.