Limit Theorems for Stochastic Processes and Random Fields via Projective Conditions

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Many statistical analyses in the social and natural sciences don't take proper account of shifts of distributions. This variability can be seen in the shift in the distribution in stock market, in the income distribution, in the electoral map, in the levels of a river or of the waves. All the data arising from these phenomena are considered to be non-stationary. Non-stationary data, as a rule, cannot be easily modeled and studied. The results obtained by many of the existing models may be spurious. They may indicate a relationship between two variables, where one does not exist. In order to apply consistent, reliable results, the rigorous mathematical tools have to be constructed for non-stationary data. This research has double scope. First, it will contribute to a better understanding and modeling of the non-stationarity phenomena and, in addition, will build new, reliable tools for their forecast and their long term behavior.An important technique for establishing limit theorems for nonstationary sequences and random fields is to approximate them with well understood structures, such as martingales and ortho-martingales. Motivated by the study of asymptotic properties of nonstationary evolutions the theory of nonstationary martingale approximations will be developed. This theory is fundamental for obtaining new projective criteria for nonstationary stochastic processes and for random fields that ensure maximal inequalities and asymptotic results, including the conditional functional central limit theorem and limit theorems started at a point for multi-indexed evolutions. The processes to be studied are very general. They include random fields with long memory and therefore they are useful to model data arising from many applied fields, such as data from economics, social sciences or engineering.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.