It is well-known that the appropriately normalized Vervaat process asymptotically behaves like one half times the squared empirical process. During the talk we shall give a complete description of the strong and weak asymptotic behaviour of the Lp-distance between these two processes. In particular, the law of the iterated logarithm, the ``other'' law of the iterated logarithm, weak convergence, etc. for the aforementioned distance will be discussed in detail.