A contribution to large deviations for heavy-tailed random sums  

作　　者：苏淳 唐启鹤 江涛 

出　　处：《Science China Mathematics》2001年第4期438-444,共7页中国科学：数学（英文版）

基　　金：This work was supported by the National Natural Science Foundation of China (Grant No. 10071081) .

摘　　要：In this paper we consider the large deviations for random sums $S(t) = \sum _{i = t}^{N(t)} X_i ,t \geqslant 0$ , whereX n,n?1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t?0 is a process of non-negative integer-valued random variables, independent ofX n,n?1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t?0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.In this paper we consider the large deviations for random sums S(t)=∑N(t)i=1Xi, t≥0, where ｛Xn, n≥1｝ are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, and ｛N(t), t≥0｝ is a process of non-negative integer-valued random variables, independent of ｛Xn, n≥1｝. Under the assumption that the tail of F is of Pareto's type (regularly or extended regularly varying), we investigate what reasonable condition can be given on ｛N(t), t≥0｝ under which precise large deviation for S(t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.

关 键 词：(extended) regular variation extreme value theory large deviations renewal counting process renewal risk model subexponential distributions 

分 类 号：O211.5[理学—概率论与数理统计]