Precise Large Deviations of Random Sums in Presence of Negative Dependence and Consistent Variation Yiqing Chen a, * , Kam C. Yuen b , Kai W. Ng b a Department of Mathematical Sciences, the University of Liverpool, Liverpool, L69 7ZL, UK b Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong May 23, 2010 Abstract The study of precise large deviations for random sums is an important topic in insurance and finance. In this paper, we extend recent results of Tang (2006) and Liu (2009) to random sums in various situations. In particular, we establish a precise large deviation result for a nonstandard renewal risk model in which innovations, modelled as real-valued random variables, are negatively dependent with common consistently- varying-tailed distribution, and their inter-arrival times are also negatively dependent. Keywords : Consistent variation; counting process; extended lower/upper negative de- pendence; precise large deviation; uniformity. 1 Introduction We say that a distribution F on (-∞, ∞) has a consistently-varying tail, written as F ∈C , if F (x)=1 - F (x) > 0 for all x and lim y&1 lim inf x→∞ F (xy) F (x) = 1, or, equivalently, lim y%1 lim sup x→∞ F (xy) F (x) =1. Clearly, the class C covers the famous class R of distributions with regularly-varying tails in the sense that the relation lim x→∞ F (xy) F (x) = y -α holds for some α ≥ 0 and all y> 0. According to Ebrahimi and Ghosh (1981) and Block et al. (1982), random variables {X k ,k =1, 2,...} are said to be lower negatively dependent (LND) if for each n =1, 2,... * Correspondence: Yiqing Chen, Department of Mathematical Sciences, the University of Liverpool, Liv- erpool, L69 7ZL, UK; E-mail: yiqing.chen@liv.ac.uk; Tel.: 44-151-794-4749; Fax: 44-151-794-4754 1