The limit points in $\bar{R^d}$ of averages of i.i.d. random variables

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.”