Statistics & Probability Letters 52 (2001) 91 – 100 Large deviations for heavy-tailed random sums in compound renewal model Qihe Tang a; b; ∗ , Chun Su a , Tao Jiang a , Jinsong Zhang b; c a Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China b The Lab of Financial Engineering Research, Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China c Guotai J&A Securities Co. Ltd., Shenzhen, Guangdong 518002, People’s Republic of China Received June 2000; received in revised form September 2000 Abstract In the present paper we investigate the precise large deviations for heavy-tailed random sums. First, we obtain a result which improves the relative result in Kl uppelberg and Mikosch (J. Appl. Probab. 34 (1997) 293). Then we introduce a more realistic risk model than classical ones, named the compound renewal model, and establish the precise large deviations in this model. c  2001 Elsevier Science B.V. All rights reserved MSC: primary 60F10; 60F05; 60G50; secondary 60K10; 62P05 Keywords: (Compound) Renewal risk model; (Extended) Regular variation; Large deviations; Renewal counting process 1. Introduction Throughout, {X n ;n¿1} denotes a sequence of i.i.d., non-negative rv’s with a common df F and a nite expectation ; independent of a process of non-negative, integer-valued rv’s {N (t );t ¿0}: We assume that (t )= EN (t )¡∞ for all t ¿0 but (t ) →∞: All limit relations, unless explicitly stated, are for t →∞; or consequently for (t ) →∞: S (t )= N (t )  i =1 X i (1) denotes randomly indexed sums (random sums) with (t )= ES (t )= (t )¡∞, where and throughout, by convention, ∑ 0 i =1 X i =0:S n = ∑ n i =1 X i as usual. All rv’s are supposed not to degenerate to zero. * Correspondence address: Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China. E-mail address: crane@mail.ustc.edu.cn (Q. Tang). 0167-7152/01/$ - see front matter c  2001 Elsevier Science B.V. All rights reserved PII: S0167-7152(00)00231-5