"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.