TITLE: Slight variation on law of the iterated logarithm
QUESTION [6 upvotes]: Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability one,$$\limsup_{t \to \infty} {{M_t - m_t}\over{\sqrt{t \log \log t}}} = r?$$If so, what is $r$?

REPLY [4 votes]: You can use the fact that $B_t-m_t$ is a reflected Brownian motion (see e.g. Revuz-Yor, Chapter VI, Theorem 2.3). I think it shouldn't be difficult to show that
$$
\limsup_{t\to\infty} \frac{M_t-m_t}{\sqrt{t\log\log t}} = \limsup_{t\to\infty} \frac{B_t-m_t}{\sqrt{t\log\log t}}.
$$