Abstract
Given any closed subset $C$ of $\bar{R^d}$, containing a pair of antipodal points at $\infty$, there is a sequence of independent and identically distributed random variables $\mathbf{X}_i}$ such that the set of limit points (in the topology of $\bar{R^d}$ of $\{(\mathbf{X}_1 + \cdots + \mathbf{X}_t)/t\}_{t \geq 1}$ equals $C$. Here $\bar{R^d}$ is the compact space gotten by “adjoining the sphere, $S^{d -1}\infty$ at infinity.”
Citation
K. Bruce Erickson. "The limit points in $\bar{R^d}$ of averages of i.i.d. random variables." Ann. Probab. 28 (1) 498 - 510, January 2000. https://doi.org/10.1214/aop/1019160128
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