Chung's law of the iterated logarithm for iterated Brownian motion (Q1917690)

Let \(X\) and \(Y\) be independent, standard Brownian motions. It is shown that the iterated Brownian motion \(\{Z(t), t\geq 0\}\), where \(Z(t)=X(|Y(t)|)\), obeys \[ \liminf_{t\to\infty} {(\log\log t)^{3/4}\over t^{1/4}}\sup_{0\leq s\leq t}|Z(s)|={\pi^{3/2}e^{3/4}\over 2^{11/4}} \] almost surely. This result is reminiscent of \textit{K. L. Chung's} law of the iterated logarithm [Trans. Am. Math. Soc. 64, 205-233 (1948; Zbl 0032.17102)].