On probability of high extremes for product of two independent Gaussian stationary processes

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• 作者：Vladimir I. Piterbarg ; Alexander Zhdanov
• 关键词：Gaussian processes ; Gaussian chaos ; High extremes probabilities ; Double sum method ; Primary 60G15 ; Secondary 60K30 ; 60K40 ; 60G70
• 刊名：Extremes
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：18
• 期：1
• 页码：99-108
• 全文大小：182 KB
• 参考文献：1. Hashorva, E, Korshunov, DA, Piterbarg, VI (2013) On extremal behavior of gaussian chaos. Dokl. Math. 88: pp. 566-568 CrossRef
  2. Hashorva, E., Korshunov, D.A., Piterbarg, V.I.: Extremal Behavior of Gaussian Chaos via Geometric Approach. Extremes, submitted (2014)
  3. Piterbarg, VI (1996) Asymptotic Methods in the Theory of Gaussian Processes and Fields. In: Transl. Math. Monographs, Vol. 148. AMS, Providence, RI
  4. Piterbarg, VI (1994) High excursions for nonstationary generalized chi-square processes. Stoch. Process Appl. 53: pp. 307-337 CrossRef
  5. Piterbarg, V, Stamatovic, S (2004) Limit Theorem for High Level a-Upcrossings by χ-Process. Theor. Probab. Appl. 48: pp. 734-741 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics
  Statistics
  Quality Control, Reliability, Safety and Risk
  Civil Engineering
  Hydrogeology
  Environmental Management
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer U.S.
• ISSN：1572-915X

文摘

Let X(t), Y(t), t?, be two independent zero-mean stationary Gaussian processes, whose covariance functions are such that r i (t)=1?|t| a i +o(|t| a i ) as \(t\rightarrow 0\) , with 0a i ?, i=1,2 and both of the functions are less than one for non-zero t. We derive for any p the exact asymptotic behavior of the probability \(P(\max _{t\in \lbrack 0,p]}X(t)Y(t)>u)\) as \(u\rightarrow \infty \) . We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order.