We use excursion theory and the ergodic theorem to present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. A simplified version of our method also proves, in a paragraph, the classical theorem of Darling and Erdős (1956).
"An Extreme-Value Analysis of the LIL for Brownian Motion." Electron. Commun. Probab. 10 196 - 206, 2005. https://doi.org/10.1214/ECP.v10-1154