Cramér-Type Moderate Deviation for Studentized Compound Poisson Sum

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• 作者：Bing-Yi Jing ; Qiying Wang ; Wang Zhou
• 关键词：Cramér large deviation ; Self ; normalized compound Poisson sum ; Studentized compound Poisson sum ; 62E20 ; 60G50
• 刊名：Journal of Theoretical Probability
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：28
• 期：4
• 页码：1556-1570
• 全文大小：429 KB
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• 作者单位：Bing-Yi Jing (1)
  Qiying Wang (2)
  Wang Zhou (3)
  
  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
  2. School of Mathematics and Statistics, University of Sydney, Sydney, Australia
  3. Department of Statistics and Applied Probability, National University of Singapore, Singapore, Singapore
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

Let \(N\) be a Poisson distributed random variable (r.v.) with parameter \(\lambda \). Let \(\{X, X_i, i \ge 1\}\) be a sequence of i.i.d. r.v’s that are independent of \(N\). Set \(S_N=\sum _{j=1}^NX_j\) and \(V_N^2=\sum _{j=1}^NX_j^2\). Assume that \(0<\mu =EX<\infty \) and \(EX^4 < \infty \). In this paper, it is proved that \(P(S_N-\lambda \mu \ge x V_N ) / \{1- \Phi (x)\} \rightarrow 1\) uniformly in \(x \in [0, o(\lambda ^{1/6}))\), as \(\lambda \rightarrow \infty \). Keywords Cramér large deviation Self-normalized compound Poisson sum Studentized compound Poisson sum