  We study the asymptotic behaviour of stochastic processes that are generated
by sums of partial sums of i.i.d. random variables and their renewals. We
conclude that these processes cannot converge weakly to any nondegenerate
random element of the space $D[0,1]$. On the other hand we show that their
properly normalized integrals as Vervaat-type stochastic processes converge
weakly to a squared Wiener process. Moreover, we also deal with the asymptotic
behaviour of the deviations of these processes, the so-called Vervaat-error
type processes.
