Precise large deviations for dependent random variables with heavy tails
Introduction
Throughout, let be non-identically distributed real-valued random variables. Their dfs are satisfying and , respectively, with finite means and . Let denote the partial sums. In this paper, we are interested in the asymptotic expressions of uniformly for all where under the assumption that are heavy-tailed. For the classical results, we refer the reader to Nagaev (1969), Nagaev (1979), Heyde, 1967, Heyde, 1968, Cline and Hsing (1991), Vinogradov (1994), Rozovski (1993), Mikosch and Nagaev (1998), Tang et al. (2001), and Ng et al. (2004).
In risk theory, heavy-tailed distributions are often used to model large claim amounts. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. A df is said to belong to class if for any . A df is said to belong to class if for any . A df is said to belong to class for some if for any . A df is said to belong to class if Some closely related discussions of the class can be found in Stadtmüller and Trautner (1979), Bingham et al. (1987), Cline (1994), and Cline and Samorodnitsky (1994). As for the applications of the class in queueing systems and ruin theory, we refer the reader to Schlegel (1998) and Jelenković and Lazar (1999). For the above heavy-tailed distribution classes, it is well known that the following relationships hold for any : Cline and Samorodnitsky (1994) constructed some examples to show that these inclusion relationships strictly hold. For more details about the classes of heavy-tailed distributions, we refer the reader to Embrechts et al. (1997), Meerschaert and Scheffler (2001), and Tang and Tsitsiashvili (2003), among others.
Recently, for practical reasons, precise large deviations of dependent random variables have received a remarkable amount of attention. Wang and Tang (2004) and Geluk and Ng (2006) both showed that the negatively associated (NA) structure has no effect on the asymptotic behavior of partial sums. Tang (2006) focused on precise large deviations of sums of ND random variables and obtained the following proposition.
Proposition 1.1 Let be ND with common distribution and mean 0 satisfyingIf there exists some such that , then for each fixed , the relationholds uniformly for all , i.e.,Condition(1.1)is unnecessary when are mutually independent.
It is natural to ask whether (1.2) can be generalized to the case of non-identically distributed random variables following a more general dependence structure than the ND structure. In other words, to what extent can precise large deviations remain insensitive to dependence structure? This paper finds out that the asymptotic behavior of precise large deviations for non-identically distributed random variables is insensitive to the END structure, which is an extension of the ND structure. More interestingly, it is proven that condition (1.1) can be weakened to
Definition 1.1 We call random variables END if there exists a constant such that both and hold for each and all .
It is worth mentioning that Paulauskas and Skučaitė (2003) obtained precise large deviations for independent and non-identically distributed real-valued random variables by using a Kolmogrov-type law of large numbers. They assumed that are independent real-valued random variables with dfs and , respectively. Furthermore, they assumed the following:
Assumption 1.1 For some , it holds that .
Assumption 1.2 For some , hold uniformly for all .
Proposition 1.2 Let be independent real-valued random variables with . IfAssumption 1.1, Assumption 1.2hold, where for with finite mean, thenfor all real-valued sequences satisfying
Motivated by the idea of assuming non-identical distribution in Paulauskas and Skučaitė (2003), this paper first extends the work of Tang (2006) by relaxing the assumptions to non-identically distributed END random variables (see Theorem 2.1, Theorem 2.2). In the proofs of those main results, a partition is used to overcome the difficulties resulting from the dependence structure and the two-sided support of . The precise large deviation results are then extended to the random sum , where is a nonnegative and integer-valued random variable independent of (see Theorem 2.3). We refer the reader to Kass and Tang (2003) and Wang and Tang (2004) for motivations of this study.
The main structure of this paper is as follows. Section 2 shows the main results. Section 3 proves the main results after preparing several lemmas. Section 4 provides two examples of sequences of END random variables.
Section snippets
Main results
For a random variable , we write , . We first state precise large deviation results for partial sums.
Theorem 2.1 Let be END with for . IfAssumption 1.2holds, satisfies(1.3)with a finite mean, and there exists some such that and , then for any relation(1.2)holds uniformly for all as .
Theorem 2.2 Let be END with for . IfAssumption 1.2holds, where satisfies(1.3)with a finite mean, then for large enough relation
Several lemmas
Let and be two infinitesimals satisfying We write if ; if ; if , if , and if both.
For a df , let We call the upper Matuszewska index of the df . Obviously, if for , then . For other properties related to the upper Matuszewska index, see Cline and Samorodnitsky (1994), Tang and Tsitsiashvili (2003)
Examples
In this section, we provide two examples to illustrate that the extended negative dependence indeed allows a wide range of dependence structures.
Example 4.1 If and are independent of each other, where is possibly valued at and is a sequence of mutually independent random variables. Then the random variables are END. In fact, for any and such that both (1.4), (1.5) hold trivially. Additionally, for any
Acknowledgments
The author would like to thank the editor, an associate editor and the referees for their constructive and insightful comments and suggestions that greatly improved the paper. The author thanks an anonymous referee who kindly pointed out a mistake in the proofs of the main results. The author also thanks Professor Yijun Hu for the discussions. This work was supported by the National Natural Science Foundation of China (10571139) and the Youth Foundation of Hubei Province Department of Education
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