Abstract
Let $S_n = X_1 + \cdots + X_n$ where $(X_n)$ is a sequence of 0-mean i.i.d. random vectors in a $B$-space such that $P(\|X_n\| > t) \leq CP(|X_0| > t)$ for some random variable $X_0 \in L_p$. We show that $S_n/n^{1/p} \rightarrow 0$ in $L_p$ iff $B$ is $p$-stable $(1 \leq p < 2)$.
Citation
Andrzej Korzeniowski. "On Marcinkiewicz SLLN in Banach Spaces." Ann. Probab. 12 (1) 279 - 280, February, 1984. https://doi.org/10.1214/aop/1176993393
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