On Complete Convergence in Mean of Normed Sums of Independent Random Elements in Banach Spaces

Abstract

For a sequence of random elements {T _n , n ≥ 1} in a real separable Banach space 𝒳, we study the notion of T _n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T _n converging completely to 0 and (ii) T _n converging to 0 in mean of order p. When 𝒳 is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements {V _n , n ≥ 1} in 𝒳 and a sequence of constants b _n  → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.

Keywords:

• Complete convergence
• Complete convergence in mean
• Convergence in mean
• Normed sums of independent random elements
• Rademacher type p Banach space
• Real separable Banach space

Mathematics Subject Classification:

• 60B11
• 60B12
• 60F15
• 60F25

ACKNOWLEDGMENTS

The authors are grateful to Professors Nguyen Duy Tien (Viet Nam National University, Ha Noi) and Nguyen Van Quang (Vinh University, Nghe An Province, Vietnam) for their interest in our work and for some helpful and important remarks.