A Strong Law of Large Numbers under Sublinear Expectations

Abstract

We consider a sequence of i.i.d. random variables $\{{\xi }_k\}$under a sublinear expectation $\mathbb{E}={\sup }_{P\in \Theta }{E}_P$. We first give a new proof to the fact that, under each $P\in \Theta$, any cluster point of the empirical averages ${\bar{\xi }}_n=({\xi }_1+\cdots +{\xi }_n)/n$ lies in $[\underline{\mu },\bar{\mu }]$ with $\underline{\mu }=-\mathbb{E}[-{\xi }_1],\bar{\mu }=\mathbb{E}[{\xi }_1]$. Then, we consider sublinear expectations on a Polish space $\Omega$, and show that for each constant $\mu \in [\underline{\mu },\bar{\mu }]$, there exists a probability $P_{\mu }\in \Theta$ such that \begin {eqnarray}\label {intro-a.s.} \lim_{n\rightarrow\infty}\barξ_n=μ, \ P_μ\textmd{-a.s.}, \end {eqnarray} supposing that $\Theta$ is weakly compact and $\{{\xi }_n\}\in L_{\mathbb{E}}^1(\Omega )$. Under the same conditions, we can get a generalization of (\ref {intro-a.s.}) in the product space $\Omega ={\mathbb{R}}^{\mathbb{N}}$ with $\mu \in [\underline{\mu },\bar{\mu }]$ replaced by $\Pi =\pi ({\xi }_1,\cdots ,{\xi }_d)\in [\underline{\mu },\bar{\mu }]$, where $\pi$ is a Borel measurable function on ${\mathbb{R}}^d$, $d\in \mathbb{R}$. Finally, we characterize the triviality of the tail $\sigma$-algebra of i.i.d. random variables under a sublinear expectation.