The Vervaat Process in Lp Spaces

It is well known that, asymptotically, the appropriately normalized Vervaat process behaves like one half times the squared empirical process. Considering these two processes as elements of the Lp-space, 1⩽p<∞, we give a complete description of the strong and weak asymptotic behaviour of the Lp-distance between them, and thus of the Lp-norm of the Vervaat process as well. The herein obtained results also raise a number of further mathematical and probabilistic problems which we formulate as suggestions for future consideration.