Exceptional times and invariance for dynamical 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; Universidad Costa Rica
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-005-0435-6
• 发表日期: 2006
• 页码: 383-416
• 关键词: points; sets; thin; LAW

摘要

Consider a sequence (X-i(0)}(i=1)(n) of i.i.d. random variables. Associate to each Xi (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X-i(t)}(t >= 0) (i = 1, 2,..) whose invariant distribution is the law nu of X-1 (0). Benjamini et al. (2003) introduced the dynamical walk S-n (t) = X-1 (t) +...+ X-n(t), and proved among other things that the LIL holds for n bar right arrow S-n(t) for all t. In other words. the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X-i (0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n bar right arrow S-n(t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov epsilon-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that the infinite-dimensional process t bar right arrow S-[n.](t)/root n converges weakly in D(D([0, 1])) to the Ornstein-Uhlenbeck process in C([0, 1]). For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini et al. (2003) by proving that if the X-i(0)'s are lattice, mean-zero variance-one, and possess (2 + epsilon) finite absolute moments for some epsilon > 0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest.