Precise large deviations for dependent random variables with heavy tails

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Abstract

By extending the negatively dependent (ND) structure, the paper puts forth the concept of extended negative dependence (END). The results show that the END structure has no effect on the asymptotic behavior of precise large deviations of partial sums and random sums for non-identically distributed random variables on (,+).

Introduction

Throughout, let {Xi,i1},X be non-identically distributed real-valued random variables. Their dfs are Fi(x)=P(Xix) satisfying F¯i(x)=1Fi(x),i1 and F(x), respectively, with finite means μi,i1 and μ. Let Sn=i=1nXi denote the partial sums. In this paper, we are interested in the asymptotic expressions of P(Sn>x) uniformly for all xγn where γ>0 under the assumption that {Xi,i1} 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 F is said to belong to class D if lim supxF¯(xy)F¯(x)< for any 0<y<1. A df F is said to belong to class L if limxF¯(x+y)F¯(x)=1 for any y>0. A df F is said to belong to class R(α) for some α>0 if limxF¯(xy)F¯(x)=yα for any y>0. A df F is said to belong to class C if limy1lim infxF¯(xy)F¯(x)=1,or equivalently,limy1lim supxF¯(xy)F¯(x)=1. Some closely related discussions of the class C 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 C 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 α>0: R(α)CLD.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 {Xi,i1} be ND with common distribution FC and mean 0 satisfyingxF(x)=o(F¯(x)),x.If there exists some r>1 such that E(X1)r< , then for each fixed γ>0 , the relationP(Sn>x)nF¯(x),nholds uniformly for all xγn , i.e.,limnsupxγn|P(Sn>x)nF¯(x)1|=0.Condition(1.1)is unnecessary when {Xi,i1} are mutually independent.

The definitions, properties and applications regarding NA can be found in Alam and Saxena (1981) and Joag-Dev and Proschan (1983); the definitions and properties regarding ND can be found in Lehmann (1966) and Block et al. (1982).

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 F(x)=o(F¯(x)),x+.

Definition 1.1

We call random variables {Xi,i1} END if there exists a constant M>0 such that both P(X1x1,,Xnxn)Mi=1nP(Xixi) and P(X1>x1,,Xn>xn)Mi=1nP(Xi>xi) hold for each n=1,2, and all x1,,xn.

Recall that random variables {Xi,i1} are called ND if both (1.4), (1.5) hold when M=1; they are called positively dependent (PD) if the inequalities (1.4), (1.5) hold both in the reverse direction when M=1. Obviously, an ND sequence must be an END sequence. On the other hand, for some PD sequences, it is possible to find a corresponding positive constant M such that both (1.4), (1.5) hold. Therefore, the END structure is substantially more comprehensive than the ND structure in that it can reflect not only a negative dependence structure but also a positive one, to some extent. For instance, the END random variables {Xi,i1} in Example 4.1 can be taken as ND or PD since there are no restrictions on the dependence structure between X1 and X2.

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 {Xi,i1},X are independent real-valued random variables with dfs Fi(x),i1 and F(x), respectively. Furthermore, they assumed the following:

Assumption 1.1

For some 1<q2, it holds that n=1nqE|Xn|q<.

Assumption 1.2

For some T>0, (a) limn1ni=1nF¯i(x)F¯(x)=1and(b) limn1ni=1nFi(x)F(x)=1 hold uniformly for all xT.

On the basis of these assumptions, they established the following proposition:

Proposition 1.2

Let {Xi,i1} be independent real-valued random variables with μi=0,i1 . IfAssumption 1.1, Assumption 1.2hold, where FR(α) for α>1 with finite mean, thenP(Sn>tn)nF¯(tn),nfor all real-valued sequences {tn} satisfyinglimnsupntn1<,F¯i(tn)=o(nF¯(tn)),1in, as n.

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 {Xi,i1}. The precise large deviation results are then extended to the random sum Sτ, where τ is a nonnegative and integer-valued random variable independent of {Xi,i1} (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 X, we write X+=max{X,0}, X=min{X,0}. We first state precise large deviation results for partial sums.

Theorem 2.1

Let {Xi,i1} be END with μi=0 for i1 . IfAssumption 1.2holds, FC satisfies(1.3)with a finite mean, and there exists some r>1 such that E(Xi)r<,i1 and E(X)r< , then for any γ>0 relation(1.2)holds uniformly for all xγn as n .

Theorem 2.2

Let {Xi,i1} be END with μi=0 for i1 . IfAssumption 1.2holds, where FC satisfies(1.3)with a finite mean, then for large enough γ relation

Several lemmas

Let f(x) and g(x) be two infinitesimals satisfying alim infxf(x)g(x)lim supxf(x)g(x)b. We write f(x)=O(g(x)) if b<; f(x)=o(g(x)) if b=0; f(x)g(x) if b=1, f(x)g(x) if a=1, and f(x)g(x) if both.

For a df F(x), let JF=inf{logγ(y)logy:y1}andγ(y)=lim infxF¯(xy)F¯(x). We call JF the upper Matuszewska index of the df F(x). Obviously, if FR(α) for α>0, then JF=α. 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 {Xi,i=1,2} and {Xi,i3} are independent of each other, where X1 is possibly valued at x11x12x1N and {Xi,i3} is a sequence of mutually independent random variables. Then the random variables {Xi,i1} are END. In fact, for any x1 and x2 such that P(X1x1)P(X2x2)=0orP(X1>x1)P(X2>x2)=0, both (1.4), (1.5) hold trivially. Additionally, for any x1

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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