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A Strong Law of Large Numbers under Sublinear Expectations

We consider a sequence of i.i.d. random variables $\{ξ_k\}$under a sublinear expectation $\mathbb{E}=\sup_{P\inΘ}E_P$. We first give a new proof to the fact that, under each $P\inΘ$, any cluster point of the empirical averages $\barξ_n=(ξ_1+\cdots+ξ_n)/n$ lies in $[\underlineμ, \barμ]$ with $\underlineμ=-\mathbb{E}[-ξ_1], \barμ=\mathbb{E}[ξ_1]$. Then, we consider sublinear expectations on a Polish space $Ω$, and show that for each constant $μ\in [\underlineμ,\barμ]$, there exists a probability $P_μ\inΘ$ such that \begin {eqnarray}\label {intro-a.s.} \lim_{n\rightarrow\infty}\barξ_n=μ, \ P_μ\textmd{-a.s.}, \end {eqnarray} supposing that $Θ$ is weakly compact and $\{ξ_n\}\in L^1_{\mathbb{E}}(Ω)$. Under the same conditions, we can get a generalization of (\ref {intro-a.s.}) in the product space $Ω=\mathbb{R}^{\mathbb{N}}$ with $μ\in [\underlineμ,\barμ]$ replaced by $Π=π(ξ_1, \cdots,ξ_d)\in [\underlineμ,\barμ]$, where $π$ is a Borel measurable function on $\mathbb{R}^d$, $d\in\mathbb{R}$. Finally, we characterize the triviality of the tail $σ$-algebra of i.i.d. random variables under a sublinear expectation.