Tail behavior of supremum of a random walk when Cramer＇s condition fails  

作　　者：Changjun YU Yuebao WANG 

机构地区：[1]School of Mathematical Sciences, Soochow University, Suzhou 215006, China [2]School of Sciences, Nantong University, Nantong 226019, China

出　　处：《Frontiers of Mathematics in China》2014年第2期431-453,共23页中国高等学校学术文摘·数学（英文）

基　　金：Acknowledgements The authors were grateful to the two reviewers for their valuable comments and suggestions to improve the present paper. This work was supported by the National Natural Science Foundation of China （NO. 11071182） and the Doctor Introduction Foundation of Nantong University （No. 12R066）.

摘　　要：We investigate tail behavior of the supremum of a random walk in the case that Cramer＇s condition fails, namely, the intermediate case and the heavy-tailed ease. 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.We investigate tail behavior of the supremum of a random walk in the case that Cramer＇s condition fails, namely, the intermediate case and the heavy-tailed ease. 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.

关 键 词：Random walk SUPREMUM exponential distribution class O-subexponential distribution class closure property asymptotic estimate ruin probability 

分 类 号：O159[理学—数学] V241.558[理学—基础数学]