Randomization/virtual-re-sampling methods, Changepoint detection, Short and long memory processes, Self-normalized partial sums processes, Planar random walks, Strong and weak approximations

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This PROPOSAL consists of four distinct research plans that are based on recent advances made in each, in the publications of the applicant with his collaborators. PROPOSAL (1) is concerned with exploring the nature of information contained in a sample of observations that are of interest to us, like for example incidence of a disease in epidemiology. We consider the problem of making inference when the available data is either small or big. For the case of having a relatively small sample, we propose a manifold extension of our recently established randomization technique that yields a significantly smaller magnitude of error of inference as compared to that of the classical procedures under the same conditions. Our randomization technique will provide a direct efficient alternative route to a well known computationally intensive randomization technique that is based on repeated re-sampling, that we are to avoid. When the data set is too big to be processed, or when it constitutes a random sample from a super-population, we are to use our randomization technique to base inference on significantly smaller sub-samples. We believe that the anticipated outcomes of this research will be important to theoretical and applied statisticians alike, and will also benefit the theory and practice of data processing for inference in various research fields of interest. PROPOSAL (2) is to explore the possibility of direct applications of our recent achievements in estimating the theoretical mean (average) of observations when it can exist at all, via self (random)- normalization of sums of observations of a random sample. This will be done by making simulations of interest feasible by representing the theoretical outcomes of our recent results in terms of random entities whose properties are, in principle, computable. We'll also explore the possibility of adapting the randomization techniques of PROPOSAL (1) to the context in hand. The anticipated significance of this work is similar to that of PROPOSAL (1). Our PROPOSAL (3) on random walks and Brownian motion on a spider graph, a 'picture' of a spider with a number of legs, will contribute to basic knowledge in mathematical sciences that may eventually become useful in studying complex systems in different areas of science and internet traffic models. The already ongoing studies of PROPOSAL (4) have been inspired by important papers that, in part, were motivated by the so-called transport phenomena of statistical physics, i.e., all irreversible processes of statistical nature stemming from the random continuous motion of molecules, mostly observed in fluids. We hope to have a fairly comprehensive comparative analysis of two approaches to anisotropic random walks that have been dominant in modeling transport phenomena.