Grant: Discovery Grants Program - Individual

Title of Proposal: Invariance Principles, Self-normalized Sums, Changepoint Analysis, Error Processes We start with a quotation from Bertrand Russel: "Although this may seem a paradox, all science is dominated by the idea of approximation." Invariance Principles are concerned with rates of approximation of various observable (data based) processes by theoretically recognizable ones.  Thus the former can, in principle and also in practice, be studied by the latter. While doing this one must also be aware of possible changes in the underlying processes. Indeed, the problem of abrupt changes arises in a variety of experimental and mathematical sciences. For instance, in epidemiology one may be interested in testing whether the incidence of a disease has remained constant over time, and if not in estimating the time(s) of change(s) in order to suggest possible causes.  Another example is rhythm analysis in electrocardiograms, where the use of change detection methods constitutes a part of pattern recognition analysis. Detecting abrupt changes in seismic signal processes is yet another example of what we, in general, call Changepoint Analysis.  The theory of invariance principles for Self-normalized (partial) Sums processes has recently become one of the main sources of information and inspiration in this regard.  When approximating a process by another one, one should also think of the possible errors made while doing this.  This, in turn, leads to studying the laws of fluctuation of the thus created Error Processes for independent, and also for long-range dependent sequences.  All this provides a beautiful interplay of the methods of various fields of the theory and applications of Statistics, Probability and Stochastic Processes.  In particular, the need for studying path properties of Gaussian and related stochastic processes enters and strengthens the research done in all the areas mentioned in our proposal.