I was watching the proof of LIL for Brownian motion in Durrett, and I have some questions. At the begining, it shows $\limsup_{n\to\infty}\frac{B_n}{\sqrt{2n\log n}}\leq 1$, a.s. by computing $\mathbb{P}(B_n>\sqrt{(2+\varepsilon)n\log n})$ and using Borel-Cantelli lemma 1. I could understand until this. But next, it replaces $\log n$ by $\log\log n$, and I couldn't understand it anymore.
- Why using the probablity about $\max_{t_n\leq s\leq t_{n+1}}B_s$ rather than $B_{t_n}$ like the previous proof ? It means the necessity of using reflection principle.
- How does the convergence of the series about $\max_{t_n\leq s\leq t_{n+1}}B_s$, directly lead to the (8.5.4)? and moreover, using the fact of $t\to (tf(t))^{1/2}$ is increasing?
- I can use the same step to show $\limsup_{t\to\infty}\frac{B_{t_n}}{\sqrt{2t_n\log \log t_n}}\leq 1$, a.s. by computing $\mathbb{P}(B_{t_n}>\sqrt{2(1+\varepsilon)^2t_n\log\log t_n})$, choosing $t_n=\alpha^n, \alpha>1$, but how to generalize from $t_n$ to $t$ ? $t_n$ is just a subsequence of $t$.
The following is a part of the proof in the book, thank you!
