Abstract
Giving a generalization of Berkes and Horváth (2003), we consider the Euclidean norm of vector-valued stochastic processes, which can be approximated with a vector-valued Wiener process having a linear drift. The suprema of the Euclidean norm of the processes are not far away from the norm of the processes at the right most point. We also obtain an approximation for the supremum of the weighted Euclidean norm with a Wiener process.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Berkes and L. Horváth, Approximations for the maximum of stochastic processes with drift, Kybernetika, 2003 (to appear).
Y. S. Chow and A. C. Hsiung, Limiting behavior of \(\max _{j \leqslant n} S_j /j^\alpha\) and the first passage time in a random walk with positive drift, Bulletin of Institute Mathem.Acad.Sinica 4 (1976), 35–44.
Y. S. Chow, A. C. Hsiung and K. F. Yu, Limit theorems for a positively drifting process and its related first passage times, Bulletin of Institute Mathem.Acad.Sinica 8 (1980), 141–172.
M. Csörgo and P. Révész, Strong approximations in probability and statistics, Academic Press, New York, 1981.
E. Eberlein, On strong invariance principles under dependence, Ann.Probab. 14 (1986), 260–270.
U. Einmahl, Extensions of results of Komlós, Major, and Tusnády to the multivariate case, J.Multivariate Analysis 28 (1989), 20–68.
J. Komló, P. Major and G. Tusnády, An approximation of partial sums of independent r.v.'s and the sample d.f. I, Z.Wahrsch.Verw.Gebiete 32 (1975), 111–131.
J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independent r.v.'s and the sample d.f. II, Z.Wahrsch.Verw.Gebiete 34 (1976), 33–58.
J. Kuelbs and W. Philipp, Almost sure invariance principles for partial sums of mixing B-valued random variables, Ann.Probab. 8 (1980), 1003–1036.
D. Siegmund, On the asymptotic normality of one-sided stopping rules, Ann.Math.Statist. 39 (1968), 1493–1497.
H. Teicher, A classical limit theorem without invariance or reflection, Ann.Probab. 1 (1973), 702–704.
W. Vervaat, Functional central limit theorems for processes with positive drift and their inverses, Z.Wahrsch.Verw.Gebiete 23 (1972), 245–253.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aue, A., Horváth, L. Approximations for the maximum of a vector-valued stochastic process with drift. Periodica Mathematica Hungarica 47, 1–15 (2003). https://doi.org/10.1023/B:MAHU.0000010807.67937.19
Issue Date:
DOI: https://doi.org/10.1023/B:MAHU.0000010807.67937.19