I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf\_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup\_{r \in [0,u]}|X\_r|=\frac{\pi}{2\sqrt{2}}\text{ a.s.}$$



Chung's law of the iterated logarithm for Brownian motion
A detailed proof with weakened conditions is given by Pakshirajan in [in a 1959 paper.](https://doi.org/10.1137%2F1104036)



> 
> In the present work the results of K. L. Chung (1948) concerning the
> maximum partial sums of sequences of independent random variables are
> obtained for a weaker condition. The method employed in the proof is
> analogous to the one used by Chung with the difference that, instead
> of Esseen’s approximations involving third moments, we use Berry’s
> approximations involving only second moments.
> 
> 
> 


