论文标题
根据标准期望的强大规律
A Strong Law of Large Numbers under Sublinear Expectations
论文作者
论文摘要
我们考虑了一系列I.I.D.随机变量$ \ {ξ_k\} $在sublinear期望$ \ mathbb {e} = \ sup_ {p \inθ} e_p $。我们首先给出了一个新的证明，即在每个$ p \inθ$下，经验的任何集群点平均$ \barξ_n=（ξ_1+\ cdots+cdots+ξ_n）/n $均为$ [\unidesllineμ，\barμ] $，带有$ \useverlineμ= - \ mathbb = - \ mathbbbb {e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e} \barμ= \ mathbb {e} [ξ_1] $。然后，我们考虑在波兰空间$ω$上的倍率期望，并表明，对于每个常数$μ\ in [\ usewissline，\barμ] $，存在概率$p_μ\inθ$，以便\ begin {eqnarray} \ label} \ label {into-a.s} \p_μ\ textmd {-a.s。}，\ end {eqnarray}，假设$θ$是弱紧密的，$ \ {ξ_n\} \ in l^1 _ {\ Mathbb {e}}}（ω）（ω）$。在相同的条件下，我们可以在产品空间中获得（\ ref {Into-a.s。}）的概括$ω= \ m athbb {r}^{\ mathbb {n}} $，并用$μ\ in [\useverlineμ，\barμ] $替换为$ fy $μ [\unsuessμ，\barμ] $，其中$π$是$ \ mathbb {r}^d $，$ d \ in \ mathbb {r} $上的borel可测量功能。最后，我们表征了I.I.D的尾巴$σ$ -Algebra的微不足道。在标准性期望下的随机变量。
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.
