Tail behavior of supremum of a random walk when Cramér’s condition fails

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• 作者：Changjun Yu (1) (2)
  Yuebao Wang (1)
  
• 关键词：Random walk ; supremum ; exponential distribution class ; O ; subexponential distribution class ; closure property ; asymptotic estimate ; ruin probability ; 60E05
• 刊名：Frontiers of Mathematics in China
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：9
• 期：2
• 页码：431-453
• 全文大小：217 KB
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• 作者单位：Changjun Yu (1) (2)
  Yuebao Wang (1)
  
  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China
  2. School of Sciences, Nantong University, Nantong, 226019, China
  
• ISSN：1673-3576

文摘

We investigate tail behavior of the supremum of a random walk in the case that Cramér’s condition fails, namely, the intermediate case and the heavy-tailed case. When the integrated distribution of the increment of the random walk belongs to the intersection of exponential distribution class and O-subexponential distribution class, under some other suitable conditions, we obtain some asymptotic estimates for the tail probability of the supremum and prove that the distribution of the supremum also belongs to the same distribution class. The obtained results generalize some corresponding results of N. Veraverbeke. Finally, these results are applied to renewal risk model, and asymptotic estimates for the ruin probability are presented.