Complete convergence theorems for weighted row sums from arrays of random elements in Rademacher type p and martingale type p Banach spaces

Abstract

In this correspondence, complete convergence theorems are established for weighted row sums $\{{\sum }_{i=1}^j{a}_{n,i}{V}_{n,i},1\le j\le k_n,n\ge 1\}$ from arrays $\{V_{n,i},1\le i\le k_n,n\ge 1,k_n\to \infty \}$ of random elements taking values in real separable Rademacher type $p (1\le p\le 2)$ Banach spaces as well as real separable martingale type $p (1\le p\le 2)$ Banach spaces. It is assumed that $\sup \{E\Vert V_{n,i}{\Vert }^p:1\le i\le k_n,n\ge 1\}<\infty .$ A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included.

Keywords:

• Complete convergence
• array of Banach space valued random elements
• weighted row sums
• Rademacher type p Banach space
• martingale type p Banach space
• rowwise independent
• martingale differences

AMS classification:

• Primary 60F15
• 60B12
• Secondary 60B11

Additional information

Funding

The research of T.-C. Hu was supported by the Ministry of Science and Technology, R.O.C. (MOST 107-2118-M-007-004-MY2). The research of A. Volodin was partially supported by the Natural Sciences and Engineering Research Council of Canada.