## Loglog estimates of Brown motion

Hi, let ${B(t), t\in R}$ be a Brown motion, then $$\varlimsup_{t\downarrow 0}\frac{B(t)}{\sqrt{2t\log\log(1/t)}} = 1$$ almost surely in the sense of Wiener measure. I find the result in a Chinese book, but no detailed proof is presented. So could someone provide some available sources online? Thanks a lot.

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Besides missing a factor 2 in the square-root term, you my find this easily by googling "law of the iterated logarithm". One good source is the Brownian Motion book by Moerters and Peres, people.bath.ac.uk/maspm/book.pdf(you find the result on page 119) – Stephan Sturm Sep 28 at 14:07
The law of the iterated logarithm doesn't apply directly. Use time inversion first. – Douglas Zare Sep 28 at 14:20
@Stephan Sturm: Thank you very much for you useful comment. I have find that the law of iterated logarithm also holds for fractional brown motion in the reference. Ehm, this is very nice ! – Wang Ming Sep 29 at 6:58