We prove a strong convergence theorem for Banach space valued random variables. One corollary of this result establishes necessary and sufficient conditions for the law of the iterated logarithm (LIL) in the Banach space setting. We also prove an exact generalization of the Hartman-Wintner law of the iterated logarithm provided the random variables involved take values in a real separable Hilbert space or some other Banach space with smooth norm.
"A Strong Convergence Theorem for Banach Space Valued Random Variables." Ann. Probab. 4 (5) 744 - 771, October, 1976. https://doi.org/10.1214/aop/1176995982