Approximations for the maximum of a vector-valued stochastic process with drift (Q1882117)

The central limit theorem for partial sums with drift is strongly related to the first passage times studied by \textit{D. Siegmund} [Ann. Math. Stat. 39, 1493--1497 (1968; Zbl 0279.62020)]. The central limit theorem and the law of iterated logarithm were extended by \textit{W. Vervaat} to functional limit theorems [Z. Warscheinlichkeitstheorie Verw. Geb. 23, 245--253 (1972; Zbl 0238.60018)]. The central limit theorem for the maximum of partial sums with positive means was established by \textit{H. Teicher} [Ann. Probab. 1, 702--704 (1973; Zbl 0262.60013)], and the law of iterated logarithm is due to \textit{Y. S. Chow} and \textit{A. C. Hsiung} [Bull. Inst. Math. Acad. Sin. 4, 35--44 (1976; Zbl 0368.60063)]. Similar results were obtained by \textit{Y. S. Chow, A. C. Hsiung} and \textit{K. F. Yu} to partial sums which can be observed only with errors [ibid. 8, 141--172 (1980; Zbl 0441.60075)]. \textit{I. Berkes} and \textit{L. Horváth} studied in their paper [Approximations for the maximum of stochastic processes with drift (to appear in Kybernetica)] approximations for the maxima of partial sums of a univariate sequence \(\{X_i\}\) of independent, identically distributed random variables with positive mean and finite positive variance. Giving a generalization of the results of Berkes and Horváth now the authors consider the Euclidian norm of vector-valued stochastic processes, which can be approximated with a vector-valued Wiener process having a linear drift. The suprema of the Euclidian norm of the process are not far away from the norm of the processes at the right most point. They also obtain an approximation for the supremum of the weighted Euclidian norm with a Wiener process. The references contain 12 bibliographical hints.