Two Lil-type results for the Brownian bridge

The necessary and sufficient condition for a function to be upper class relative to a Brownian motion process was obtained by T. Sirao and T. Nisida (1952, Nagoya Math. J. 4, 97–101) and a related result on first passage distribution was obtained by P. K. Sen and M. J. Wichura (in press, Estimation of First Crossing Time Distribution for Brownian Motion Processes Relative to Upper Class Boundaries). Corresponding results for the Brownian bridge are discussed.