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.
Citation
Uwe Einmahl. Jim Kuelbs. "Cluster sets for partial sums and partial sum processes." Ann. Probab. 42 (3) 1121 - 1160, May 2014. https://doi.org/10.1214/12-AOP827
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