On the $(p,q)$-type Strong Law of Large Numbers for Sequences of Independent Random Variables

Li, Qi, and Rosalsky (Trans. Amer. Math. Soc., 2016) introduced a refinement of the Marcinkiewicz--Zygmund strong law of large numbers (SLLN), so-called the $(p,q)$-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 $1\le p<2,q\ge 1$. This paper gives a complete solution to open problems raised by Li, Qi, and Rosalsky by providing the necessary and sufficient conditions for the $(p,q)$-type SLLN for the cases where $0<q\le p<1$ and $0<q<1\le p<2$. We consider random variables taking values in a real separable Banach space $\mathbf{B}$, but the results are new even when $\mathbf{B}$ is the real line. Furthermore, the conditions for a sequence of random variables $\left\{X_n, n \ge 1\right\}$ satisfying the $(p, q)$-type SLLN are shown to provide an exact characterization of stable type $p$ Banach spaces.