A Hilbert space law of the iterated logarithm is proved which generalizes Kolmogorov's law for bounded random variables and which generalizes results of Teicher for unbounded random variables. The result for identically distributed random vectors is a consequence. The key idea is the requirement of the convergence of the average of the covariance operators.
"On the Multivariate Law of the Iterated Logarithm." Ann. Probab. 7 (6) 980 - 988, December, 1979. https://doi.org/10.1214/aop/1176994891