A new proof of the Hartman-Wintner law of the iterated logarithm is given. The main new ingredient is a simple exponential inequality. The same method gives a new, simpler proof of a basic result of Kuelbs on the LIL in the Banach space setting.
"A New Proof of the Hartman-Wintner Law of the Iterated Logarithm." Ann. Probab. 11 (2) 270 - 276, May, 1983. https://doi.org/10.1214/aop/1176993596