Cluster sets for partial sums and partial sum processes

Let X,X1,X2,. be i.i.d. mean zero random vectors with values in a separable Banach space B, Sn=X1+.+Xn for n.1, and assume {cn:n.1} is a suitably regular sequence of constants. Furthermore, let S(n)(t), 0.t.1 be the corresponding linearly interpolated partial sum processes. We study the cluster sets A=C({Sn/cn}) and A=C({S(n)(.)/cn}). In particular, A and A are shown to be nonrandom, and we derive criteria when elements in B and continuous functions f:[0,1].B belong to A and A, respectively. When B=Rd we refine our clustering criteria to show both A and A are compact, symmetric, and star-like, and also obtain both upper and lower bound sets for A. When the coordinates of X in Rd are independent random variables, we are able to represent A in terms of A and the classical Strassen set K, and, except for degenerate cases, show A is strictly larger than the lower bound set whenever d.2. In addition, we show that for any compact, symmetric, star-like subset A of Rd, there exists an X such that the corresponding functional cluster set A is always the lower bound subset. If d=2, then additional refinements identify A as a subset of {(x1g1,x2g2):(x1,x2).A,g1,g2.K}, which is the functional cluster set obtained when the coordinates are assumed to be independent.