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

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 thei...

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Bibliographic Details
Published in:arXiv.org 2004-01
Main Authors: Csáki, Endre, Csörgő, Miklós, Rychlik, Zdzisław, Steinebach, Josef
Format: Article
Language:English
Subjects:
Asymptotic properties
Convergence
Random variables
Stochastic models
Stochastic processes
Sums
Online Access:Get full text
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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.
ISSN:2331-8422