Precise large deviations for sums of random variables with consistently varying tails

Abstract

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation µ > 0. Under the assumption that the tail probability F (x) = 1 − F (x) is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N (t) , where N (t) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

Large deviations for random walks under subexponentiality: The big-jump domain

“…The next theorem is due to A. Nagaev in the global case with regularly varying F ; see [14], Theorem 8.6.2 or Ng et al [35]. In the local regularly varying case, it goes at least back to Pinelis [37].…”

An Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems

“…One could also mention here some further efforts aimed at extending the conditions of Theorem A that ensure the validity of (4), see e.g. [4,5].…”

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.…”

“…Moreover, throughout this section, we let (t) . To state our results, we will need the following assumption, which was used by Ng et al (2004), and is satisfied for many common counting processes, for example, the renewal counting process and the Cox process.…”