On the (p,q)$(p,q)$‐type strong law of large numbers for sequences of independent random variables

Abstract

introduced a refinement of the Marcinkiewicz-Zygmund strong law of large numbers (SLLN), the so-called (𝑝, 𝑞)-type SLLN, where 0 < 𝑝 < 2 and 𝑞 > 0.They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: 0 < 𝑝 < 1, 𝑞 > 𝑝, and 1 ≤ 𝑝 < 2, 𝑞 ≥ 1. Results for the case where 0 < 𝑞 ≤ 𝑝 < 1 and 0 < 𝑞 < 1 ≤ 𝑝 < 2 remain open problems. This paper gives a complete solution to these problems. We consider random variables taking values in a real separable Banach space 𝐁, but the results are new even when 𝐁 is the real line. Furthermore, the conditions for a sequence of random variables {𝑋 𝑛 , 𝑛 ≥ 1} satisfying the (𝑝, 𝑞)-type SLLN are shown to provide an exact characterization of stable type 𝑝 Banach spaces. K E Y W O R D Scomplete convergence in mean, (𝑝, 𝑞)-type strong law of large numbers, real separable Banach space, stable type 𝑝 Banach space, strong law of large numbers 402