Some mean convergence theorems for weighted sums of Banach space valued random elements

Abstract

In this correspondence, we investigate mean convergence of order p for the weighted sums of Banach space valued random elements under a suitable (compactly) uniformly integrable condition with or without a geometric condition placed on the Banach space.

Keywords:

• Mean convergence
• Banach space valued random element
• martingale difference sequence
• Rademacher type p Banach space
• martingale type p Banach space

Acknowledgments

The authors are grateful to three referees for carefully reading the manuscript and for offering many substantial suggestions which resulted in an improved presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of M. Ordóñez Cabrera was partially supported by the Plan Andaluz de Investigacion de la Junta de Andalucia FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The research of A. Volodin was partially funded by the program of support of the Mathematical Center of the Volga Region Federal District (Project 075-02-2020-1478).