Some "LIM INF" Results for Increments of a Wiener Process

• Published in: The Annals of probability Vol. 17; no. 3; pp. 1063 - 1082
• Main Authors: Hanson, D. L., Russo, Ralph P.
• Format: Journal Article
• Language: English
• Published: Hayward, CA Institute of Mathematical Statistics 01-07-1989; The Institute of Mathematical Statistics
• Subjects: 60F15; 60G15; 60G17; Approximation; Exact sciences and technology; Increments of a Wiener process; Integers; Lebesgue measures; Limit theorems; Mathematical theorems; Mathematics; Partial sums; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Topological theorems; Wiener process
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/aop/1176991257

Let W(t) for$0 \leq t < \infty$be a standard Wiener process, suppose$0 < a_T \leq T$for$T > 0$, and let d(T, t) = {2t[log(T/t) + log log t ]}1/2. Quantities such as$\lim, \inf_{T \rightarrow \infty} \sup_{a_T \leq t \leq T} \frac{W(T) - W(T - t)}{d(T,t)},$
$\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq t \leq T - a_T\\0 \leq s \leq a_T}} \frac{|W(t + s) - W(t)|}{d(t + a_T, a_T)}$and$\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq u < v \leq T\\a_T \leq v - u}} \frac{|W(v) - W(u)|}{d(v, v - u)}$are investigated.