Cluster sets for partial sums and partial sum processes

Let $X, X_1, X_2,\ldots$ be i.i.d. mean zero random vectors with values in a separable Banach space $B$, $S_n=X_1+\cdots +X_n$ for $n \ge 1$, and assume $\{c_n: n\ge 1\}$ is a suitably regular sequence of constants. Furthermore, let $S_{(n)}(t), 0 \le t \le 1,$ be the corresponding linearly interpolated partial sum processes. We study the cluster sets $A= C(\{S_n/c_n\})$ and $\mathcal{A}=C(\{S_{(n)}(\cdot )/c_n\})$. In particular, $A$ and $\mathcal{A}$ are shown to be non-random and we derive criteria when elements in $B$ and continuous functions $f: [0,1] \to B$ belong to $A$ and $\mathcal{A}$, respectively. When $B= \mathbb{R}^d$ we refine our clustering criteria to show both $A$ and $\mathcal{A}$ are compact, symmetric, and star-like, and also obtain both upper and lower bound sets for $\mathcal{A}$. When the coordinates of $X$ in $\mathbb{R}^d$ are independent random variables we are able to represent $\mathcal{A}$ in terms of $A$ and the classical Strassen set $\mathcal{K}$, and except for degenerate cases show $\mathcal{A}$ is strictly larger than the lower bound set whenever $d \ge 2$. In addition, we show that for any compact, symmetric, star-like subset $A$ of $\mathbb{R}^d$, there exists an $X$ such that the corresponding functional cluster set $\mathcal{A}$ is always the lower bound subset. If $d=2$, then additional refinements identify $\mathcal{A}$ as a subset of $\{ (x_1g_1, x_2g_2): (x_1,x_2) \in A, g_1,g_2 \in \mathcal{K}\}$, which is the functional cluster set obtained when the coordinates are assumed to be independent.