The laws of the iterated logarithm for two kinds of PP statistics

Motivated by recent work on exploratory Projection Pursuit (PP), we define the smoothing PP Kolmogorov-Smirnov statistics and the smoothing PP Crámer-von Mises statistics based on one-dimensional projections. We demonstrate the Laws of the Iterated Logarithm (LIL) for the PP Kolmogorov-Smirnov statistics, the PP Crámer-von Mises statistics hold. The conditions and results of the LIL in this paper are same as those of the LIL in one-dimensional empirical processes.