Which properties of a random sequence are dynamically sensitive?

• 成果类型: Article
• 署名作者: Benjamini, I; Häggström, O; Peres, Y; Steif, JE
• 署名单位: University of Gothenburg; University of California System; University of California Berkeley; Hebrew University of Jerusalem; University System of Georgia; Georgia Institute of Technology; Chalmers University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1-34
• 关键词: brownian-motion; random-walks; capacity

摘要

Consider a sequence of i.i.d. random variables, where each variable is refreshed (i.e., replaced by an independent variable with the same law) independently, according to a Poisson clock. At any fixed time t, the resulting sequence has the same law as at time 0, but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated. We prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm, so these laws are dynamically stable. However, there are times at which run lengths are exceptionally long, that is, run tests are dynamically sensitive. We obtain a multifractal analysis of exceptional times for run lengths and for prediction. In particular, starting from an i.i.d. sequence of unbiased random bits, the random set of times t where alphalog(2)(n) bits among the first n bits can be predicted from their predecessors, has Hausdorff dimension 1 - alpha a.s. Finally, we consider simple random walk in the lattice Z(d), and prove that transience is dynamically stable for d greater than or equal to 5, and dynamically sensitive for d = 3, 4. Moreover, for d = 3, 4, the nonempty random set of exceptional times t where the walk is recurrent has Hausdorff dimension (4 - d)/2 a.s.