"On Vervaat and Vervaat-error type processes for partial sums and renewals"

Endre Csáki, Miklós Csörgö, Zdzislaw Rychlik and Josef Steinebach

Summary: 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.

Keywords: Partial sums; renewals; Vervaat and Vervaat-error type processes; Wiener process; strong and weak approximations; weak convergence.

AMS 2000 subject classification: 60F17 60F05 60F15.