A generalization of Strassen's functional LIL
Abstract
Let X1,X2, . . . be a sequence of i.i.d. mean zero random variables and let Sn the sum of the first n random variables. We show that whenever lim supn |Sn|/cn is finite with probability one and the normalizing sequence cn is sufficiently regular, the corresponding normalized partial sum process sequence is relatively compact in C[0, 1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.
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