The limit points of the strong law of large numbers under the sub-linear expectations

Abstract

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables relative to a regular sub-linear expectation $(\Omega ,\mathcal{H},\hat{\mathbb{E}})$ with upper mean $\overline{\mu }$, lower mean $\underline{\mu }$, and finite Choquet expectation or ${\lim }_{c\to \infty }\hat{\mathbb{E}}[(|X_1|-c{)}^{+}]=0$. Then for any Borel-measurable function $\phi (x_1,\dots ,x_d)$ on ${\mathbb{R}}^d$ or continuous function $\phi (x_1,x_2,\dots )$ on ${\mathbb{R}}^{\mathbb{N}}$, ${\sum }_{i=1}^n{X}_i/n$ converges to $\underline{\mu }\wedge \phi (X_1,X_2,\dots )\wedge \overline{\mu }$ with upper capacity $1$. The set of limit points of ${\sum }_{i=1}^n{X}_i/n$ is the whole mean interval $[\underline{\mu },\overline{\mu }]$ with upper capacity $1$, as is any random subset of the mean interval whose boundaries are continuous functions or finite-dimensional Borel-measurable functions of $(X_1,X_2,\dots )$, also with upper capacity $1$.