Title: A generalization of Strassen's functional LIL.
Let X_1, X_2, ... be a sequence of i.i.d. mean zero random variables and let
S_n denote the sum of the first n random variables. We show that whenever we have with probability one, limsup |S_n|/c_n = alpha_0 is finite as n tends to infinity for a regular normalizing sequence c_n, 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. Our proof is based on a new strong invariance principle for sums of i.i.d. random variables with possibly infinite variance.
