Weighted approximations, self-normalization and long-range dependance

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Results and methods of strong and weak approximations have become an integral part of the theory and applications of probability and statistics in the last 40 years. Recent contributions in this area are mainly concerned with weighted approximations of stochastic processes based on i.i.d. observations (cf., e.g., M.Csörgö and L.Horváth, Weighted Approximations in Probability and Statistics, Wiley 1993). We construct new tools for use in weighted approximations of additive processes in different metrics. The long term objective of my research remains to be creating a unified theory of weighted asymptotics of various stochastic processes in different metrics under optimal conditions. This goal has already been achieved for partial sum processes, U-statistics type processes, sums of sequential ranks, empirical processes of observations, ranks and sequential ranks in my earlier papers. Moreover, in [2] and [10], we succeeded in extending Donsker's theorem via weighted sup-norm and Lp-approximations in probability for self-normalized partial sum processes of i.i.d. random variables in the domain of attraction of the normal law (DAN). This in particular extends the theory of invariance principles to partial sums whose variance is not necessarily finite. The construction of these stochastic processes and also that of the earlier ones that I study is such that statistical tests can be based on them for detecting a possible change in their distribution. The problem of abrupt changes arises in areas like quality control, epidemiology, seismic signal processing, or in the study of archaeological sites.