Normal approximation by Stein’s method under sublinear expectations

By Yongsheng Song. Stochastic Processes and their Applications, 2017

51 citations recorded by the source corpus.

Peng (2008)(\cite{P08b}) proved the Central Limit Theorem under a sublinear expectation: \textit{Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d random variables under a sublinear 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 Lipschitz function $\varphi$, \[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|=0,\] 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 this CLT 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 positive constant $C_{\alpha, G}$ depending on $\alpha, \underline{\sigma}$ and $\overline{\sigma}$ such that \[\sup_{|\varphi|_{Lip}\le1}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|\leq C_{\alpha,G}\frac{\hat{\mathbf{E}}[|X_1|^{2+\alpha}]}{n^{\frac{\alpha}{2}}},\] 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 \[G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overline{\sigma}^2a^+-\underline{\sigma}^2a^-), \ a\in \mathbb{R}.\]}

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f5bf915096ded1c8d67b7dd9e9e47a8674225ed7

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In Finance

  1. Yongsheng Song
  2. Shige Peng
  3. Qi-Man Shao
  4. Xiao Fang
  5. Marcel Nutz
  6. Jianfeng Zhang
  7. Mingshang Hu
  8. Li-Xin Zhang

In Management Science and Operations Research

  1. Yongsheng Song
  2. Qi-Man Shao
  3. Xiao Fang
  4. Shige Peng
  5. Li-Xin Zhang
  6. Marcel Nutz
  7. Jianfeng Zhang
  8. Mingshang Hu

In Statistics and Probability

  1. Yongsheng Song
  2. Xiao Fang
  3. Qi-Man Shao
  4. Shige Peng
  5. Louis H. Y. Chen
  6. Sourav Chatterjee
  7. Li-Xin Zhang
  8. Giovanni Peccati

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