Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

Abstract

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {X ij , j ≥ 1}, i = 1, . . . , k. In this paper we investigate large deviations for both partial sums S (k; n 1 . . , k, are counting processes for the claim number. The obtained results extend some related classical results.

Sums of Pairwise Quasi-Asymptotically Independent Random Variables with Consistent Variation

“…This class has recently been applied to the study of precise large deviations by several authors including Ng et al [20] , Tang [27] , Wang and Wang [33] , and Robert and Segers [25] . An obvious property of the class is that, for all y > 0,…”

Precise Large Deviations for Sums of Independent Random Variables with Consistently Varying Tails

“…Class is slightly larger than the ERV class. Ng et al (2004) studied the asymptotic relations of precise large deviations on the class , see Wang and Wang (2007) and Konstantinides and Loukissas (2010) for more details.…”

Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

“…For more details on regularly varying tails and extended regularly varying tails, see Klüppelberg and Mikosch (1997) or Tang et al (2001). Recently, Ng et al (2004) considered a subclass of heavy-tailed distributions slightly larger than the ERV class, called class C. We restate their definition as follows.…”

“…Some earlier work on large deviations can be found, for example, in Nagaev (1969) and Heyde (1967). Nagaev (1973Nagaev ( ), (1979 studied the large deviation probabilities (1.1) for the claims with regularly varying tails. Cline and Hsing (1991) and Klüppelberg and Mikosch (1997) extended the results to the so-called ERV (extended regularly varying) class.…”