Weak a-uniform integrability of sequences of random elements

This thesis consists of five chapters. The first chapter is devoted to the introduction. In the second chapter, some basic concepts of functional analysis and real analysis are recalled. In the third chapter, the concepts of weak A-statistical convergence and weak A-strong summability are introduced in normed spaces and the equivalence of these concepts for bounded sequences is given. It is showed that weak A-statistical convergence coincide with A-statistical convergence in weak topology. After the concept of weak A-uniform integrability is defined, it is shown that a sequence is weakly A-strongly summable if and only if it is weakly A-statistically convergent and weakly A-uniformly integrable in normed spaces. In the fourth chapter, the concepts of A-statistical convergence and A-strong summability are generalized for sequences of random elements in topological vector spaces by using the Pettis integral. A relationship between these more general concepts and the concepts is given in the third chapter is obtained. Also we introduce a new versiyon of uniform integrability that is called A-bounded uniformly Pettis integrable for the sequences of random elements. It is showed that this new concept is generalization of Pettis uniform integrability and A-compact uniform Bochner integrability. Some theorems and results related to the concept of Pettis A-bounded uniform integrability are given. Finally, in the last chapter, the analysis of the obtained results is given.