Local asymptotic classes for the successive primitives of Brownian motion

• 成果类型: Article
• 署名作者: Lachal, A
• 署名单位: Universite Claude Bernard Lyon 1
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1023481108
• 发表日期: 1997
• 页码: 1712-1734
• 关键词: gaussian-processes; spline

摘要

Let (B(t))(t greater than or equal to 0) be the linear Brownian motion starting at 0, and set X-n(t) = (1/n!) integral(0)(t)(t - s)(n) dB(s). Watanabe stated a law of the iterated logarithm for the process (X-1(t))(t greater than or equal to 0), among other things, This paper proposes an elementary proof of this fact, which can be extended to the general case n greater than or equal to 1. Next, we study the local asymptotic classes (upper and lower) of the (n + 1)-dimensional process U-n = (B, X-1,..., X-n) near zero and infinity, and the results obtained are extended to the case where B is the d-dimensional Brownian motion.