Single and multiple averages along subsequences

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This proposal's main purpose is to investigate an area of mathematics which grew out of the science of interpreting data of large number of measurements. The basic problem is this: under normal circumstances, when people have to make repeated measurements of some data to draw statistical conclusions, they try to make the measurements at regular intervals, like once every second. The existing theories usually assume that the measurements are taken _exactly_ at every second. In reality, measurements are made at every second only approximately. The proposal's investigations are aimed to look at these possible random perturbations, and try to develop tools to help interpret the obtained data. One of the main motivation for these investigations is the fact that random perturbations may result in a completely different set of data from those one could obtain under the ideal circumstances. As it may be guessed, many of the problems we face are difficult, and to solve these problems we resort to tools used in several different branches of mathematics.