Cluster sets for partial sums and partial sum processes

Abstract

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 $n\ge1$, and assume $\{c_{n}:n\ge1\}$ is a suitably regular sequence of constants. Furthermore, let $S_{(n)}(t)$, $0\le t\le1$ 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 nonrandom, 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 $d\ge2$. 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 $\{(x_{1}g_{1},x_{2}g_{2}):(x_{1},x_{2})\in A,g_{1},g_{2}\in\mathcal{K}\}$, which is the functional cluster set obtained when the coordinates are assumed to be independent.