论文标题
在$（p，q）$ - 键入大量的强大定律，用于独立随机变量的序列
On the $(p,q)$-type Strong Law of Large Numbers for Sequences of Independent Random Variables
论文作者
论文摘要
Li，Qi和Rosalsky（Trans。Amer。Math。Soc。，2016年）介绍了Marcinkiewicz-Zygmund大型法律（SLLN）的改进，所谓的$（p，q）$ - 类型 - 类型SLLN，其中$ 0 <p <2 $和$ q> 0 $。他们为这两种情况下的新类型SLL获得了一组必要和充分的条件：$ 0 <p <1 $，$ q> p $和$ 1 \ le p <2，q \ ge 1 $。本文通过为$ 0 <q \ le p <1 $和$ 0 <q <q <q <1 \ le p <2 $提供了$（p，q）$ - 类型SLL的必要条件，从而为Li，Qi和Rosalsky提出的开放问题提供了完整的解决方案。我们考虑随机变量在真正可分开的Banach Space $ \ Mathbf {B} $中取值，但是即使$ \ Mathbf {B} $是真实行，结果也是新的。此外，一系列随机变量的条件$ \ weft \ {x_n，n \ ge 1 \ right \} $满足$（p，q）$ - 类型slln的条件可提供稳定的类型$ p $ p $ p $ p $ banach空间的精确表征。
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.
