The author studies the almost sure behaviour of the increments of the partially observed empirical process and derives some functional laws of the iterated logarithm for this process. Application to nonparametric density estimation is presented.

The Hausdorff dimension of the set generated by exceptional oscillations of the uniform empirical process is studied. We correct a former result obtained by Deheuvels and Mason (1995, Ann. Probab. 23, 355-387).