This paper studies Strong Laws of Large Numbers for Banach space-valued random variables which are subject to the Banach space analog of orthogonality called weak orthogonality.
"Strong Laws of Large Numbers for Weakly Orthogonal Sequences of Banach Space-Valued Random Variables." Ann. Probab. 2 (5) 918 - 925, October, 1974. https://doi.org/10.1214/aop/1176996557