Weak max-sum equivalence for dependent heavy-tailed random variables

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1 DOI /s Lithuanian Mathematical Journal, Vol. 56, No. 1, January, 2016, pp Wea max-sum equivalence for dependent heavy-tailed random variables Lina Dindienė a and Remigijus Leipus a,b,1 a Faculty of Mathematics and Informatics, Vilnius University, Naugarduo str. 24, LT Vilnius, Lithuania b Institute of Mathematics and Informatics, Vilnius University, Aademijos str. 4, LT Vilnius, Lithuania lina_dindiene@yahoo.com; remigijus.leipus@mif.vu.lt Received November 28, 2014; revised June 13, 2015 Abstract. We consider real-valued random variables X 1,...,X n with corresponding distributions F 1,...,F n such that X 1,...,X n admit some dependence structure and n 1 F F n belongs to the class of dominatedly varyingtailed distributions. We establish wea equivalence relations among PS n > x, Pmax{X 1,...,X n } > x, Pmax{S 1,...,S n } >x, and n F x as x,wheres := X X. Some copula-based examples illustrate the results. MSC: 60E05, 60G70, 62E20 Keywords: wea max-sum equivalence, heavy tails, dominatedly varying tails, negative dependence, copula 1 Introduction and main result Let X 1,...,X n be real-valued random variables r.v.s with corresponding distributions F 1,...,F n. Denote F x :=1 F x and S := X X for =1,...,n. Li and Tang [12] and Yang et al. [18] investigated the wea equivalence relations among the quantities PS n > x, Pmax{X 1,...,X n } > x, Pmax{S 1,...,S n } > x, and n F x as x, considering, respectively, the independent or pairwise negative dependent see the definition below random variables X 1,...,X n together with the condition that their maximum belongs to a specific class of heavytailed distributions. In this paper, we consider the class of dominatedly varying-tailed distributions, denoted by D, and dependence structure given further in 1.4, which covers a wide range of negative and some positive dependent structures. Recall that a distribution function d.f. F x =1 F x belongs to the class D if lim sup F xy/f x < for any or for some 0 <y<1. It is well nown that F D if and only if F xy L F := lim lim inf y 1 x F x > 0. 1 The author is supported by a grant No. MIP from the Research Council of Lithuania /16/ c 2016 Springer Science+Business Media New Yor 49

2 50 L. Dindienė and R. Leipus Recall some related dependence structures. Random variables X 1,...,X n are said to be upper extended negatively dependent UEND if there exists a positive constant M such that, for all x 1,...,x n, PX 1 >x 1,..., X n >x n M n PX i >x i ; 1.1 they are said to be lower extended negatively dependent LEND if there exists a positive constant M such that, for all x 1,...,x n, n PX 1 x 1,..., X n x n M PX i x i ; 1.2 and they are said to be extended negatively dependent END if they are both UEND and LEND see [13]. When M =1in 1.1 and1.2, the r.v.s X 1,...,X n are said to be upper negatively dependent UND and lower negatively dependent LND, respectively, and they are said to be negatively dependent ND if 1.1 and 1.2 both hold with M =1;see[4, 6, 17]. Random variables X 1,...,X n are called pairwise upper extended negatively dependent pairwise UEND if PX i >x i,x j >x j MPX i >x i PX j >x j 1.3 for all x i,x j R, i j, i, j {1,...,n}, andsomem > 0. Similarly, the related positive dependence structures can be introduced. Denote the d.f. x :=n 1 F 1 x+ + F n x and assume that x > 0 for all x. Introduce the following condition: PX >x, X l >x=o1 x, x, 1.4 or, equivalently, 1<ln PX >x, X l >x=o1 x for all, l =1,...,n, <l. 1.5 Random variables satisfying 1.4 allow a wide range of dependence structures. In particular, they cover the pairwise ND r.v.s and even some positive dependence structures see Section 3. They also include dependent r.v.s characterized by the condition PX >x, X l >x=o1 F x+f l x with 1 <l n, 1.6 which, in the case where X 1,...,X n are all nonnegative, generate the pairwise quasi-asymptotically independence structure; see [5]. The dependence structure 1.5 strictly contains the structure 1.6. To see this, tae the trivial example n =3, X 1 = X 2 with distribution F 1 = F 2 = F, and independent r.v. X 3 with distribution F 3 such that F x =of 3 x. ThenX 1,X 2,X 3 do not satisfy 1.6 but satisfy 1.5. Note that under some stronger dependence conditions, related equivalence results for subexponential r.v.s were established by Gelu and Tang [8] andjiangetal.[11]. When X 1,...,X n are real-valued and identically distributed r.v.s, the dependence structure 1.5 coincides with the bivariate upper tail independence BUTI structure. Note that the BUTI is strictly larger than the UEND structure. To see this, consider two positive r.v.s ξ 1 and ξ 2 with the joint tail probability Pξ 1 >x, ξ 2 >y= 1, x 0, y 0. x 1y 11 + x + y Such a pair ξ 1,ξ 2 is bivariate upper tail independent but not UEND see Example 3.1 in [14]. The goal of the paper is to prove some wea max-sum equivalence relations under condition 1.4 and provide some concrete dependence structures satisfying 1.4.

3 Wea max-sum equivalence for dependent heavy-tailed random variables 51 Let G n x =1 G n x :=Pmax{X 1,...,X n } >x, S n := max{s 1,...,S n },andt n := X X + n,wherex + := max{x, 0}. All the limit relationships further hold for x tending to. For two positive functions ax and bx, we write ax bx if lim sup ax/bx 1 and ax bx if lim inf ax/bx 1. The main result of the paper is the following theorem. Theorem 1. Let r.v.s X 1,...,X n satisfy condition 1.4. If D or, equivalently, G n D, then If, in addition, x =o x, then PS n >x PT n >x L 1 n x. 1.7 PS n >x PS n >x L Hn n x. 1.8 Here, L Hn = L Gn and n x G n x. 1.9 Remar 1. Yang et al. [18] proved that for pairwise ND r.v.s X 1,...,X n such that G n D,wehave and, under the condition G n x =og n x, PS n >x PT n >x L 1 G n G n x 1.10 PS n >x PS n >x L Gn G n x Since pairwise ND r.v.s satisfy condition 1.4, Theorem 1 generalizesthe result in [18]; moreover see Proposition 1, the constant L Gn in 1.10and1.11 can be replaced by L Hn. Remar 2. Clearly, in the case F C D, =1,...,n,whereC denotes the consistently varying-tailed class of distributions, characterized by F xy lim lim sup y 1 x F x =1, we have C and thus L Hn =1in Theorem 1. However, there may exist some more relaxed conditions implying C; see, for example, a method in Theorem 1.1 of [11]. In the case of identically distributed random variables, we obtain the following corollary. Corollary 1. Let assumptions of Theorem 1 hold, and let X 1,...,X n be identically distributed with common distribution F. Then relations 1.7 and 1.8 hold with L Hn = L Gn = L F and x =F x. In Section 2, we formulate an auxiliary proposition and prove the main theorem. In Section 3, we illustrate the result using some copula-based dependence structures. We start with the following useful proposition. 2 Proof of main result Proposition 1. Assume that condition 1.4 holds. Then G n x n x, and therefore L Gn = L Hn. Lith. Math. J., 561:49 59, 2016.

4 52 L. Dindienė and R. Leipus Proof. We have On the other hand, n G n x =P {X >x} PX >x. 2.1 G n x PX >x PX >x, X l >x <ln From 1.4 and it follows that x is positive for all x if and only if G n x > 0 is positive for all x. Then and G n x lim inf n x 1 lim sup lim sup G nx n x 1 1<ln PX >x,x l >x n x =1, implying G n x n x and, thus, L Gn = L Hn. Proof of Theorem 1. Relations 1.9 hold by Proposition 1, implying the equivalence of D and G n D. We first show the upper bound 1.7. For any 0 <v<1and x>0, write n { n { PT n >x P X + > 1 vx} + P T n >x, X + 1 vx} We have by D that n 1 vx + P T n >x, =: I 1 v, x+i 2 v, x. n { X + i > x n }, n { X + 1 vx} I 1 v, x lim lim sup v 0 x L 1 n x = L 1 vx lim lim sup =1. v 0 x x As for I 2 v, x, wehave I 2 v, x P T n >x, X i + > x n n, { X + 1 vx} P T n X i + >vx, X i + > x n j=1 j i P X j + > vx n,x+ i > vx. n n P j=1 j i { X j + > vx },X i + > x n 1 n

5 Wea max-sum equivalence for dependent heavy-tailed random variables 53 Hence, by 1.4 and the assumption D we obtain lim sup I 2 v, x L 1 n x L lim sup i j PX i >vx/n,x j >vx/n n vx/n lim sup vx/n x =0. Therefore, lim sup x PT n >x L 1 lim lim sup n x v 0 x I 1 v, x L 1 n x + lim lim sup v 0 x To obtain the lower bound, note that, for any v>0and x>0, n { PS n >x P S n >x, X > 1 + vx } P S n >x, X > 1 + vx 1i<jn I 2 v, x L 1 n x =1. P S n >x, X i > 1 + vx, X j > 1 + vx =: I 3 v, x I 4 v, x. 2.3 Here, by 1.4, For I 3 v, x, wehave I 4 v, x PX i >x, X j >x=o x i<jn Here, I 3 v, x P S n X > vx, X > 1 + vx PSn X > vx+f 1 + vx 1 = n 1 + vx PS n X vx =:I 31 v, x I 32 v, x. 2.5 I 31 v, x lim lim inf = v 0 x L Hn n x For the term I 32 v, x, since D,wehave I 32 v, x = P X i vx i n 2 v n 1 x n P i = o1 v n 1 x { X i v } n 1 x = o x. 2.7 Lith. Math. J., 561:49 59, 2016.

6 54 L. Dindienė and R. Leipus Hence, by , lim inf x PS n >x L Hn n x I 31 v, x lim lim inf v 0 x L Hn n x lim I 32 v, x lim sup v 0 x L Hn n x I 4 v, x lim lim sup v 0 x L Hn n x =1. The proof is completed. 3 Modeling negative dependence structures with copulas In this section, we discuss some copula-based examples of dependence structures satisfying 1.4. It is clear that any pairwise ND or pairwise UEND r.v.s X 1,...,X n satisfy 1.4. Moreover, some positive dependent structures can be constructed as well. 3.1 Generalized FGM copula Consider the class of generalized Farlie Gumbel Morgenstern GFGM copulas given by the formula Q GFGM u 1,...,u n = n u i 1+ 1<ln θ l 1 u 1 u l m 3.1 with >0, m {0, 1, 2,...}, andθ l taing values from a corresponding admissible region. Obviously, if θ l are all nonpositive and tae values from a corresponding admissible region, then Q GFGM u 1,...,u n u 1...u n, that is, we obtain the LND structure. The following particular cases of 3.1 are well nown: If m =0, then we get the independence copula. If m =1and =1, then we get the classical multivariate FGM copula Q FGM u 1,...,u n = n u i 1+ 1<ln θ l 1 u 1 u l, which was introduced in [7, 9] and[16] in the casen =2. This copula was widely investigatedand used in practice. The well-nown limitation to FGM copula is that it does not allow the modeling of high dependencies. For example, if n =2, then the admissible region for the parameter θ 12 is [ 1, 1], and the correlation ρ between the corresponding uniformly distributed random variables is ρ = θ 12 /3; thus, the range for correlation ρ is [ 1/3, 1/3]. If m =1, n =2, and >0, then we get the copula introduced by Huang and Kotz [10]. It was shown that the admissible range of θ 12 is min{1, 2 } θ 12 1 and the correlation ρ between the corresponding uniformly distributed random variables is ρ =3θ ; thus, the range for correlation ρ is min{1, 2 } ρ If m 1, n =2, and >0, then we get the copula introduced by Berizadeh et al. [3]. They have shown that the admissible range of θ 12 is min{1, m 2 1 } θ 12 m 1 and the correlation between the corresponding uniformly distributed random variables is given by the formula 1 ρ = Q GFGM u, vdu dv 3=12 m m Γ + 1Γ2/ 2 θ12. Γ +1+2/

7 Wea max-sum equivalence for dependent heavy-tailed random variables 55 Because of the wea dependence generated by the FGM family, many authors considered the modifications of this class. Examples of modified FGM copula can be found in [1, 2], among others. The finding of the admissible region for parameters θ l in 3.1 is technical, although straightforward, tas. Essentially, it requires the verification that the corresponding copula density if exists q GFGM u 1,...,u n = n Q GFGM u 1,...,u n / u 1... u n is nonnegative for all u 1,...,u n. In the case of copula 3.1 with m =1, q GFGM u 1,...,u n =1+ 1<ln θ l u u l, and these conditions can be obtained by considering the 2 n cases for u =0or 1, =1,...,n, and verifying that q GFGM u 1,...,u n 0. For example, if m =1and n =3, then these conditions are the following: and 1+ 2 θ 0, θ l 1+θ 0, θ l { θ 1 1+ if θ > 1, 1 θ 1 1+ if θ 1, { 1 θ 1 1+ if θ > 1, θ 1 1+ if θ 1, 1 <l 3, for >1 1 <l 3, for 0 < 1 with θ := θ 12 + θ 13 + θ 23. Note that any pair of variables X 1,...,X n, lined by copula 3.1, satisfies PX x, X l y = F x,f l y, l,where Q GFGM l Q GFGM l u, v =uv 1+θ l 1 u 1 v m. 3.2 Obviously, 3.2 impliesq GFGM l u, v uv, <l, whenever θ l are all nonpositive. Hence, the generalized FGM copula 3.1 provides a pairwise negative dependence structure if θ l 0, 1 <ln. The following proposition shows that this copula also captures the pairwise UEND structure, which contains some positive dependence structures too. Proposition 2. Let the distribution of X 1,...,X n be generated by copula in 3.1. Then where C l := 1 + max{, 1} θ l +1 m 1. PX >x, X l >y C l F xf l y, 3.3 Proof. For every <l,by3.2wehavethat PX x, X l y =F xf l y [ 1+θ l 1 F x 1 F l y] m. Hence, PX >x, X l >y=1 F x F l y+px x, X l y =1 F x F l y+f xf l y 1+θ l 1 F x 1 Fl y m = F x+f l y 1+ 1 F x F l y+f xf l y m m 1+ θ i i l F x i Fl y i Lith. Math. J., 561:49 59, 2016.

8 56 L. Dindienė and R. Leipus = F xf l y+ 1 F x F l y+f xf l y m m θ i i l F x i Fl y i, where F x :=1 F x. Using the inequality 1 u max{, 1}1 u, u [0, 1], weget m PX >x, X l >y F xf l y+f xf l y m θ l i i 1+max{, 1} θ l +1 m 1 F xf l y. Obviously, if θ l in 3.1 are all nonnegative, then X 1,...,X n are both lower positive dependent and pairwise positive dependent. By 3.3 this is also a pairwise UEND structure. 3.2 Ali Mihail Haq copula Consider the copula of the form Q AMH u 1,...,u n = u 1 u n, 1 θ<1, θ1 u 1 1 u n and let PX 1 x 1,..., X n x n =Q AMH F 1 x 1,...,F n x n. Then, for l, and hence PX x, X l y = F xf l y 1 θf xf l y, PX >x, X l >y=1 F x F l y+ F xf l y 1 θf xf l y F xf l y 3.5 if 1 θ 0. In the case 0 <θ<1, wehave PX x, X l y 1 1 θ F xf l y, 3.6 PX >x, X l >y 1 1 θ F xf l y. 3.7 Inequality 3.6 is obvious. In order to show 3.7, it suffices to verify that 1 u v + uv 1 θ1 u1 v 1 u1 v, 0 u, v 1, 0 <θ<1. 1 θ The proof is straightforward, and we omit it. By the copula in 3.4 generates a pairwise ND structure if 1 θ 0 and a pairwise END structure which is also lower positive dependent and pairwise positive dependent if 0 <θ<1.

9 Wea max-sum equivalence for dependent heavy-tailed random variables Fran copula Consider the copula of the form Q F u 1,...,u n = 1 θ log 1 e 1+ e θu1 θun 1 e θ 1 n 1, θ > 0, 3.8 and assume that PX 1 x 1,..., X n x n =Q F F 1 x 1,...,F n x n. ThenPX x, X l y = Q F F x,f l y, l, where In this case, the copula density is bounded: q F u, v = Q F u, v := 1 θ log 1+ e θu 1e θv 1 e θ. 1 θe θ 1e θu+v e θ 1 + e θu 1e θv 1 2 Thus, denoting the corresponding marginal densities f x, wehave PX >x, X l >y= w>x,z>y θ 1 e θ e 2θ =: c θ. q F F w,f l z f wf l zdw dz c θ F xf l y, l, that is, the Fran copula generates a pairwise UEND structure. 3.4 Clayton copula Consider the copula Q C u 1,...,u n = u θ u θ n n +1 1/θ, θ > 0, 3.9 and assume that PX 1 x 1,...,X n x n =Q C F 1 x 1,...,F n x n. Then PX x, X l y = Q C F x,f l y, where Q C u, v = u θ + v θ 1 1/θ. Note that if θ 0, then Q C u, v tends to uv, that is, we obtain the independence copula, whereas if θ, then Q C u, v tends to minu, v, that is, the comonotonicity copula. We will show that for any l and x, y R, PX >x, X l >y 1 + θf xf l y. This implies the pairwise UEND property and, hence, relation 1.4. The proof of this inequality follows from the identity PX >x,x l >y=1 F x F l y+px x, X l y and the following lemma. Lemma 1. For any u, v [0, 1] 2 and θ>0, we have u θ + v θ 1 1/θ uv + θ1 u1 v. Lith. Math. J., 561:49 59, 2016.

10 58 L. Dindienė and R. Leipus Proof. Denote, for convenience, Q θ u, v :=u θ + v θ 1 1/θ. Tae any small ɛ>0 and write Q θ u, v Q ɛ u, v = = θ ɛ θ ɛ Q t u, v t dt = θ ɛ u t + v t 1 logu t + v t 1 u t log u t v t log v t t 2 u t + v t 1 1+1/t dt Q t u, v u t + v t 1 logu t + v t 1 u t log u t v t log v t t 2 u t + v t 1 For all u, v [0, 1] 2 and t>0, we have dt. and u t + v t 1 logu t + v t 1 u t log u t v t log v t t 2 u t + v t 1 Q t u, v uv u1 v uv Bound 3.10 is due to the inequality u t/2 v t/2 2 + u t/2 v t/2 1. In order to prove 3.11, we use the following inequality: for any x 1, y 1. Denote x + y 1 logx + y 1 x log x y log y x + y 1 log x log y 3.12 fx, y :=x + y 1 logx + y 1 x log x y log y x + y 1 log x log y. Then 3.12 follows by noting that f1,y=0for any y 1 and fx, y = log x log y + y 1 x x log y +log xy x + y 1 By 3.12wehave 0, x,y 1. u t + v t 1 logu t + v t 1 u t log u t v t log v t t 2 u t + v t 1 log u log v, where, by the inequality log x x 1/ x, x 1 see [15, p. 272], log u =log 1 u 1/u 1 1/ u = 1 u u. Inequalities 3.10and3.11imply Q θ u, v Q ɛ u, v+θ ɛ1 u1 v. Taing ɛ 0, we obtain the desired inequality. Summarizing, we have the following corollary. Corollary 2. Let r.v.s X 1,...,X n have corresponding univariate distributions F 1,...,F n such that D, and let the dependence structure be generated by either of the copulas in 3.1, 3.4, 3.8, or3.9. Then the asymptotic relation 1.7 holds. If, in addition, x =o x, then1.8 also holds.

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