On dynamical Gaussian random walks

• 成果类型: Article
• 署名作者: Khoshnevisan, D; Levin, DA; Méndez-Hernández, PJ
• 署名单位: Utah System of Higher Education; University of Utah
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000001044
• 发表日期: 2005
• 页码: 1452-1478
• 关键词: probability-inequalities; Stochastic processes; random-variables; CONVERGENCE

摘要

Motivated by the recent work of Benjamini, Haggstrom, Peres and Steif [Ann. Probab. 34 (2003) 1-34] on dynamical random walks, we do the following: (i) Prove that, after a suitable normalization, the dynamical Gaussian walk converges weakly to the Ornstein-Uhlenbeck process in classical Wiener space; (ii) derive sharp tail-asymptotics for the probabilities of large deviations of the said dynamical walk; and (iii) characterize (by way of an integral test) the minimal envelope(s) for the growth-rate of the dynamical Gaussian walk. This development also implies the tail capacity-estimates of Mountford for large deviations in classical Wiener space. The results of this paper give a partial affirmative answer to the problem, raised in Benjamini, Haggstrom, Peres and Steif [Ann. Probab. 34 (2003) 1-34, Question 4], of whether there are precise connections between the OU process in classical Wiener space and dynamical random walks.