Weak A-statistical Convergence and Pettis Integrability

Speaker
Havva Uluçay
Department of Mathematics, Istanbul Technical University, Istanbul, Türkiye
Contributing Author
Mehmet Ünver
Department of Mathematics, Ankara University, Ankara, Türkiye

Summability theory, a subfield of functional analysis and topology concerned with the convergence properties of sequences, studies the assignment of limits to divergent sequences. One of its most widely investigated notions is statistical convergence, introduced independently by Fast and Steinhaus.
We study weak versions of statistical convergence and strong summability in normed and topological vector spaces. Using nonnegative regular summability matrices, we introduce the concept of weak A-statistical convergence, weak A-strong summability, and weak A-uniform integrability, and investigate the relationships among them. We show that weak A-statistical convergence coincides with A-statistical convergence in the weak topology.
Furthermore, these notions are extended to sequences of random elements in topological vector spaces via the Pettis integral. In this setting, we study Pettis A-statistical convergence and Pettis-type uniform integrability, obtaining generalizations of classical results in summability theory and functional analysis.

Keywords: Statistical convergence, Pettis integral, weak convergence.