We examine the relationship between the central limit theorem with Gaussian limit, and law of the iterated logarithm type behavior in the Banach space setting. Our results solve some problems posed by Kesten (1972) and generalize some results of Klass (1976, 1977).
J. Kuelbs. J. Zinn. "Some Results on LIL Behavior." Ann. Probab. 11 (3) 506 - 557, August, 1983. https://doi.org/10.1214/aop/1176993499