English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Peng (2008)(\cite{P08b}) proved the Central Limit Theorem under a sublinearexpectation:  \textit{Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d random variables under asublinear expectation $\hat{\mathbf{E}}$ with$\hat{\mathbf{E}}[X_1]=\hat{\mathbf{E}}[-X_1]=0$ and$\hat{\mathbf{E}}[|X_1|^3]<\infty$. Setting$W_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$, we have, for each bounded and Lipschitzfunction $\varphi$,\[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|=0,\]where $\mathcal{N}_G$ is the $G$-normal distribution with$G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]$, $a\in \mathbb{R}$.}  In this paper, we shall give an estimate of the rate of convergence of thisCLT by Stein's method under sublinear expectations:  \textit{Under the same conditions as above, there exists $\alpha\in(0,1)$depending on $\underline{\sigma}$ and $\overline{\sigma}$, and a positiveconstant $C_{\alpha, G}$ depending on $\alpha, \underline{\sigma}$ and$\overline{\sigma}$ such that\[\sup_{|\varphi|_{Lip}\le1}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|\leqC_{\alpha,G}\frac{\hat{\mathbf{E}}[|X_1|^{2+\alpha}]}{n^{\frac{\alpha}{2}}},\]where $\overline{\sigma}^2=\hat{\mathbf{E}}[X_1^2]$,$\underline{\sigma}^2=-\hat{\mathbf{E}}[-X_1^2]>0$ and $\mathcal{N}_G$ is the$G$-normal distribution with\[G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overline{\sigma}^2a^+-\underline{\sigma}^2a^-),\ a\in \mathbb{R}.\]}

Normal Approximation by Stein's Method under Sublinear Expectations