Refinement of convergence rate for the strong law of large numbers in Banach space

Abstract

Let be a sequence of independent and identically distributed B-valued random variables, and set . Let , and q>0, and put

We strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that , if and only if and

Keywords::

• Hsu–Robbins–Erdős–Spitzer–Baum–Katz complete convergence theorem
• partial sums of i.i.d. Banach space-valued random variables
• strong law of large numbers

MSC (2010) Classification::

• 60B12
• 60F15
• 60G50

Additional information

Funding

The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.