Abstract

One of the classical results of probability for sums of i.i.d. random variables is the functional LIL of Strassen (1964) who under the classical assumptions that the second moment is nite and the expectation of the underlying dis- tribution is equal to zero showed that with probability one, the sequence fS(n)= p 2n log log ng where S(n) : ! C[0; 1] is the partial sum process of order n; is relatively compact in C[0; 1] and moreover that the (random) set of limit points of this sequence is equal to a certain deterministic subset of C[0; 1] which we call the cluster set of the sequence fS(n)= p 2n log log ng. This result extends to higher dimensions and there are versions in the innite variance case where one has to use dierent normalizing sequences fcng. In the 1-dimensional case it turned out that one still gets the standard cluster set as in the Strassen LIL provided that the normalizing sequence satises some mild regularity assumptions. This is no longer the case if one looks at this problem in the multidimensional setting. The purpose of this talk is to give a survey of some recent work in this direc- tion. Among other things, we are able to determine all possible cluster sets in the independent component case. In the general case we can identify min- imal and maximal sets for the functional cluster sets in terms of the cluster sets of the normalized sums.