Laws of the iterated logarithm for time changed Brownian motion with an application to branching processes (Q1068439)

Let \(\tau_ n\), \(n\geq 1\), be a sequence of nondecreasing sequences of random variables, and B(t) be the standard Brownian motion. The law of iterated logarithm for the sequence \((2\tau_ n\) \(\log_ 2\) \(\tau_ n)^{-1/2}B(\tau_ nt)\), \(0\leq t\leq 1\), \(n\geq 1\), is proved. Here the sequence \(\tau_ n\) may increase at a geometric rate. This result is applied to various quantities associated with the Galton-Watson process.