A functional law of the iterated logarithm for time changed Brownian motion is given for stopping times that increase at a geometric rate. This result is applied to various quantities associated with a Galton-Watson process.
"Laws of the Iterated Logarithm for Time Changed Brownian Motion with an Application to Branching Processes." Ann. Probab. 13 (4) 1148 - 1156, November, 1985. https://doi.org/10.1214/aop/1176992801