Asymptotic theory for stochastic processes via martingale methods

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An important technique for establishing limit theorems for dependent sequences is to approximate them with well understood structures, such as martingales. Stationary martingale approximation is a subfield of stochastic processes that received a lot of attention in the last decade. Motivated by the study of asymptotic properties of nonstationary processes and also of stationary processes that cannot be approximated by stationary martingales, we shall develop the theory of nonstationary martingale approximations, which unites all the parts of the project. The new method will exploit blocking techniques to break the dependence and a new type of martingale construction based on blocks of variables. We shall also provide maximal inequalities, including Rosenthal-type inequalities, which are important for obtaining rates of convergence in the asymptotic results and facilitate the study of a stochastic process by approximating it, in the almost sure sense, with sums of independent normal random variables. These tools are fundamental for obtaining new projective criteria for stochastic processes that insure asymptotic results, including the conditional functional central limit theorem, limit theorems started at a point, moderate and large deviation results as well as exact representations for the tail probabilities of sums of random variables. These types of asymptotic behaviors are at the heart of probability theory with important applications to statistics and other applied fields. The proposed project is expected to provide new mathematical ideas and techniques that will shed new light on several difficult open problems for stochastic processes and will impact other fields of research as follows: The limit theorems started at a point are useful to analyze random walks in a random environment. They are also of interest to researchers working in statistical mechanics, physics, and will lead to new discoveries for some interesting intermittent maps that recently came to the attention of specialists in dynamical systems. The results will be applicable to families of Metropolis-Hastings algorithms that are essential, for instance, for Bayesian statistics. The exact asymptotic representation for the tail probabilities will facilitate the estimations of deviation probabilities that occur in a natural way in many applied areas, so for instance, in problems of insurance in the context of large claim insurance, in risk theory and finance. The results will be disseminated broadly through publications in top rated journals and presentations in national and international conferences. They will also be integrated with training of graduate students, presented in weekly seminars, and will enter the curriculum of a course on limit theory for stochastic processes. Related questions are actively being studied by several groups of researchers, in the US and in Europe, and the proposed project will contribute to strengthen international scientific exchange and collaborations.