This paper provides sharp sufficient conditions for almost sure summability of the maximal normed partial sums of m-dependent random elements in stable type p Banach spaces, complementing recent results of Li, Qi, and Rosalsky (Trans Amer Math Soc 368(1):539–561, 2016) and Thành (Math Nachr 296(1):402–423, 2023). The main theorems are new even when the underlying random elements are independent. Our proof is direct and can be extended to other dependence structures. Two illustrative examples are also presented.

中文翻译：

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Banach 空间中 m 相关随机元素的最大赋范部分和的几乎确定可求和

本文为稳定类型p Banach 空间中m相关随机元素的最大范数部分和的几乎确定的可求性提供了清晰的充分条件，补充了 Li、Qi 和 Rosalsky 的最新结果 (Trans Amer Math Soc 368(1):539 –561, 2016) 和 Thành (Math Nachr 296(1):402–423, 2023)。即使基础随机元素是独立的，主要定理也是新的。我们的证明是直接的，可以扩展到其他依赖结构。还提供了两个说明性示例。