On the random max-closure for heavy-tailed random variables

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1 DOI 0.007/s Lithuanian Mathematical Journal, Vol. 57, No. 2, April, 207, pp On the random max-closure for heavy-tailed random variables Remigijus Leipus a,b and Jonas Šiaulys a a Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT Vilnius, Lithuania b Institute of Mathematics and Informatics, Vilnius University, Akademijos str. 4, LT Vilnius, Lithuania remigijus.leipus@mif.vu.lt; jonas.siaulys@mif.vu.lt Received May 2, 206; revised December 20, 206 Abstract. In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X,X 2,..., which states that, given distributions F X,F X2,... from some class of heavy-tailed distributions, the distribution of the random maximum X η := max0,x,...,x η or random maximum S η := max0,s,...,s η belongs to the same class of heavy-tailed distributions. Here, S n = X + + X n, n, and η is a counting random variable, independent of X,X 2,... We provide the conditions for the random max-closure property in the case of classes D and L. MSC: 60E05, 62E0 Keywords: heavy tails, long tail, dominatedly varying tail, random maximum, closure property, nonidentically distributed random variables Introduction Let X,X 2,... be identically or nonidentically distributed random variables r.v.s. Let η be an integer-valued nonnegative r.v. such that Pη =0< ; we call η a counting r.v. In addition, pose that η is independent of the sequence X,X 2,... Let now S 0 := 0, S n := X + + X n, n, be the partial sums, and let S η = X + + X η be the random sum of r.v.s X,X 2,.... Similarly, let X n := max0,x,...,x n, n, X 0 := 0, and let X η := max0,x,...,x η. Finally, let S n := maxs 0,S,...,S n, n 0, andlets η := maxs 0,S,...,S η be the random maximum of sums S 0,S,S 2,.... For any real x, denote F Sη x :=PS η x = F Xη x :=PX η x = F Sη x :=PS η x = PS n xpη = n, n=0 PX n xpη = n, n=0 PS n xpη = n. n= /7/ c 207 Springer Science+Business Media New York 208

2 On the random max-closure for heavy-tailed random variables 209 In this paper, we are interested in the closure property of the random maxima X η and S η, which states that, given F X,F X2,... from some class of heavy-tailed distributions, the distributions F Xη and F Sη belong to the same class of heavy-tailed distributions. In this case, we say that corresponding random variables or distributions possess the random max-closure property. Recall that a r.v. X, together with its distribution function F X, is said to be heavy-tailed if Ee δξ = for all δ > 0. We further present definitions of three important subclasses of heavy-tailed distributions. Ad.f.F is said to belong to the class L of long-tailed d.f.s if, for every a>0, its tail F x = F x satisfies lim F x + a/f x =.Ad.f.F belongs to the class D has dominatingly varying tail if, for every a 0,, F xa/f x <. Ad.f.F, ported by [0,, belongs to the class of subexponential d.f.s S if lim F F x/f x =2,where denotes the convolution of d.f.s. If a d.f. F is ported by R, thenf S if F + x :=F x [0, x belongs to S. Lemma.3.5 and Proposition.4.4 of [0]seealso[4, ]and[4] imply that L D S L, D S, S D. It is appropriate to mention that the subject of the paper is partially motivated by the closure problem of the random convolution. In the case of independent and identically distributed i.i.d. r.v.s X, X,X 2,..., we say that some class of d.f.s H is closed with respect to the random convolution if F X H implies F Sη H. The random closure property for subexponential distributions was studied by Embrechts and Goldie see [9, Thm. 4.2] and Cline see [5, Thm. 2.3]. The random closure results for class D can be found in [7, 5]andforclassL in [, 5, 20]and[2]. Note that Xu et al. [2] and Danilenko and Šiaulys [7] considered the case where r.v.s X,X 2,... are not necessarily identically distributed. Also, we note that the closure properties for d.f.s F Sη and F Sη in the case of i.i.d. r.v.s can be derived from asymptotic formulas obtained, for instance, in [8, 3, 6, 8, 22]. Like in [5], we restrict ourselves to the classes L and D. We find minimal conditions for real-valued not necessarily identically distributed r.v.s X,X 2,... and a counting r.v. η under which the d.f.s F Xη and F Sη belong to the corresponding class. It should be noted that the case of nonrandom η is trivial since the implications F Xk L, k =,...,n F Xn L and F Xk D, k =,...,n F Xn D hold by relation PX n >x n PX k >x. The rest part of the paper is organized as follows. In Section 2, we present our results. In Section 3, we collect auxiliary lemmas. In Section 4, we give the proofs of main results. Finally, in Section 5, wepresent some examples. 2 Mainresults In this section, we formulate the main results of the paper. 2. Random max-closure for class D Theorem. Suppose that X,X 2,... are real-valued arbitrarily dependent r.v.s with corresponding d.f.s F X,F X2,..., and let η be a counting r.v., independent of X,X 2,..., such that nκ ϕn n F Xk x F Xκ x < 2. for some κ pη := n 0: Pη = n > 0 and some positive sequence ϕn, n with Eϕη [, η <. Then the following two statements are equivalent: i F Xη D, ii F Xκ D. Lith. Math. J., 572:208 22, 207.

3 20 R. Leipus and J. Šiaulys The proof of the theorem is given in Section 4. The following two corollaries follow immediately from Theorem. The first corollary deals with the case of identically distributed variables. Corollary. Let X,X 2,... be arbitrarily dependent r.v.s with common d.f. F X, and let η be a counting r.v., independent of X,X 2,..., such that Eη <. ThenF Xη D if and only if F X D. The next corollary considers the case of finite number of r.v.s. Corollary 2. Assume that m is a nonrandom integer, X,...,X m are arbitrarily dependent r.v.s, and η is a counting r.v., independent of X,...,X m, such that pη 0,,...,m. Let F Xk x F Xκ x <, k =,...,m, for some κ pη. ThenF Xη D if and only if F Xκ D. The following two results give sufficient conditions for the property F Sη D. Recall that a d.f. F belongs to the class D if and only if the upper Matuszewska index of F is finite see [2, Thm. 2..8] and discussion in Section 2 of [7]: J + F := lim y log y log F xy lim inf <. F x Remark. The first three our results, Theorem and Corollaries, 2, hold for r.v.s that can be dependent. Whereas the statements below are proved only for independent r.v.s., the structure of the presented proofs show that similar statements will be true for some mild dependence structures like NUOD negative upper orthant dependence or WUOD widely upper orthant dependence. Such dependence structures and their useful properties are described in [9] and references therein. Theorem 2. Let X,X 2... be independent real-valued r.v.s, and let η be a counting r.v. independent of X,X 2,... Assume that F Xκ D for some κ pη and Eη p+ < for some p > J + F Xκ.If condition 2. with ϕn =n is satisfied, then F Sη D. The proof of the theorem is presented in Section 4. The following corollary can be easily derived from Theorem 2. Corollary 3. Let X,X 2,... be i.i.d. r.v.s with common d.f. F X D, and let η be a counting r.v. independent of X,X 2,..., such that Eη p+ < for some p>j + F X.ThenF Sη D. 2.2 Random max-closure for class L Theorem 3. Suppose that X,X 2,... are independent real-valued r.v.s with corresponding d.f.s F X,F X2,... and η is a counting r.v. independent of X,X 2,... i Suppose that there exists κ pη such that F Xk L for all k κ, Eϕη [, η < for some positive sequence ϕn,n, and, as x, n F Xk x ϕnf Xκ x 2.2 uniformly in n κ. ThenF Xη L.

4 On the random max-closure for heavy-tailed random variables 2 ii Suppose that there exists κ pη such that F Xk x =o F Xκ x for all k<κ 2.3 and 2.2 holds uniformly in n κ. ThenF Xκ L F Xη L. Corollary 4. Let X,X 2,... be i.i.d. r.v.s with common d.f. F X, and let η be a counting r.v. independent of X,X 2,... IfEη <, thenf X L F Xη L. Consider now conditions for F Sη L. Theorem 4. Let X,X 2,... be independent real-valued r.v.s with corresponding d.f.s F X,F X2,..., and let η be a counting r.v. independent of X,X 2,... i If pη 0,,...,m m < and F Xk L for all k,...,m, thenf Sη L. ii Assume that Pη = k > 0 for all k, and let the following conditions hold: Then F Sη L. F Xk x ii. =, k F Xk x ii.2 lim n Pη >n =0 and min kn Pη = k n ii.3 Eη η>δx = o F X x for any δ 0,. Pη n Pη = n =, Corollary 5. Let X,X 2,... be i.i.d. real-valued r.v.s with common d.f. F X L, and let η be a counting r.v. independent of X,X 2,... i If η has a finite port, then F Sη L ; ii If Pη = k > 0 for all k and conditions ii.2 and ii.3 of Theorem 4 hold, then F Sη L. Corollaries 4 and 5 follow from Theorems 3 and 4, respectively. The proofs of Theorems 3 and 4 are given in Section 4. 3 Auxiliary results In this section, we present several results used in the proofs of main theorems. The first lemma is well known see, e.g., [2, Prop. 2.2.]. Lemma. For a d.f. F D and any p>j + F, there exist positive constants c and c 2 such that F u v p F v c u for all v u c 2. In addition, u p = of u for any p>j + F. The second lemma, which is an inhomogeneous case of Theorem 3 from [6], was proved in [7]. Lemma 2. Suppose that ξ,ξ 2,... are nonnegative independent r.v.s. Let, for some ν, F ξν D and Lith. Math. J., 572:208 22, 207. nν n n F ξk x F ξν x <.

5 22 R. Leipus and J. Šiaulys Then, for every p>j + F ξν, there exists a positive constant c 3 such that F ξ F ξn x F ξν x c 3 n p+ for all n ν, x 0. The third lemma is a combination of Theorem 2. and Lemma 4. from [3]. Denote ξ + := ξ ξ0. Lemma 3. Let ξ,...,ξ m be independent real-valued r.v.s such that F ξk L for all k =,...,m. Then P max k km i= ξ i x L and P max km k m m ξ i >x P ξ i >x P ξ i + >x. i= The last auxiliary lemma was proved by Embrechts and Goldie see inequality 2.2 in [9]. Lemma 4. Let F and G be two d.f.s such that F x > 0, Gx > 0, x R. Then for all x R, v R, and t>0. F Gx t F Gx max yv In this section, we present proofs of Theorems 4. i= F y t, F y yx v+t 4 Proofs i= Gy t Gy Proof of Theorem. ii i For each positive x,wehave n PX η >x= P X k >x Pη = n. 4. n= Therefore, from we get PX η >x PX κ >xpη = κ 4.2 F Xη x/2 F Xη x n n= F X k x/2pη = n. F Xκ xpη = κ Condition 2. implies that, for some c 4 > 0, forlargex, andforalln, wehave Hence, for x large enough, we have n F Xk x c 4 ϕnf Xκ x. 4.3 and the desired implication follows. F Xη x/2 F Xη x c 4Eϕη [, η Pη = κ F Xκ x/2 F Xκ x,

6 On the random max-closure for heavy-tailed random variables 23 i ii Similarly, using 4., 4.2, and 4.3, we obtain if x is sufficiently large. Hence, F Xη x/2 F Xη x F Xκ x/2pη = κ c 4 n= ϕnf X κ xpη = n F Xκ x/2 F Xκ x c 4Eϕη [0, η Pη = κ F Xη x/2 F Xη x, and the implication follows. The theorem is proved. Proof of Theorem 2. If x>0, then F Sη x = = P maxs,...,s n >x Pη = n n= n= n= P max S +,...,S n + >x Pη = n P S + n >x Pη = n, where S + k := X X+ k. Hence, by Lemma 2 we have F Sη x + P S n + >x Pη = n n<κ nκ P S κ + >x Pη <κ+c 5 n p+ Pη = nf Xκ x nκ c 5 F Xκ x κ p+ + Eη p+ 4.4 with some positive constant c 5. On the other hand, for positive x, F Sη x Pη = κp maxs,...,s κ >x Inequalities 4.4and4.5implythat Pη = κps κ >x Pη = κp X κ > 2x, X > x κ,..., X κ > x κ κ = Pη = κf Xκ 2x i= P X i > x. 4.5 κ F Sη x/2 F Sη x c 5κ p+ + Eη p+ Pη = κ = c 5κ p+ + Eη p+ Pη = κ F Xκ x/2 F Xκ 2x lim inf κ i= PX i > x/κ F Xκ x/2 F Xκ 2x <. Lith. Math. J., 572:208 22, 207.

7 24 R. Leipus and J. Šiaulys Proof of Theorem 3. i To prove that F Xη L, we have to show that F Xη x F Xη x. 4.6 Using equality 4.wehavethat,forx>0, F Xη x n<κ F Xn xpn η<κ+ n P X k >x Pη = n nκ n<κ F Xn xpn η<κ + F Xκ x nκ ϕn n F Xk x ϕlpη = l F Xκ x lκ =: s x+s 2 x. 4.7 For the lower bound, take any integer K>κ and write, using the Bonferroni inequality, F Xη x n P X k >x Pη = n+ n<κ K n P X k >x Pη = n n=κ n n n F Xk x F Xk xf Xj x Pη = n n<κ j= K K n + F Xj x Pη = n F Xk x j= n=κ κ F Xj x F Xn xpn η<κ + j= j= n<κ K F Xj x F Xκ x inf nκ ϕn n F Xk x K ϕlpη = l F Xκ x =: s 3 x+s 4 x 4.8 if x is large enough. Consider now two cases: a Pη <κ > 0; bpη <κ =0. In case a, by 4.7, 4.8, and the inequality we have a + + a m a max,..., a m b + + b m b b m F Xη x F Xη x s x max, s 2x. s 3 x s 4 x l=κ for a i 0, b i > 0 4.9

8 On the random max-closure for heavy-tailed random variables 25 From the conditions of the theorem and 4.9 we get that s x s 3 x s 2 x s 4 x n<κ F X n x Pn η<κ n<κ F X n x Pn η<κ F Xn x max =, 4.0 n<κ F Xn x F Xκ x F Xκ x nκ ϕn n lim inf inf nκ ϕn n F Xk x F Xκ x F Xk x F Xκ x Eϕη ηκ Eϕη κηk = Eϕη ηκ Eϕη κηk. 4. Hence, F Xη x F Xη x max, Eϕη ηκ Eϕη κηk for every K>κ. Letting K tend to infinity, we get inequality 4.6. Case b is obvious. ii First, we show that, under the conditions of the theorem, F Xη L F Xκ L. From 4.7wehavethat,forx>0, F Xκ x nκ ϕn n F Xk x Eϕη ηκ F Xη x F Xn xpn η<κ. F Xκ x n<κ Similarly, estimate 4.8 implies that, for K>κ andsufficientlylarge x, F Xκ x inf nκ ϕn n F Xk x Eϕη κηk F Xκ x F Xη x K j= F X j x. The last two estimates give F Xκ x F Xκ x F X η x / K j= F X j x F Xη x n<κ F X n xpn η<κ n nκ ϕn inf nκ ϕn n F Xk x F Xκ x F Xk x F Xκ x Eϕη ηκ Eϕη κηk. Lith. Math. J., 572:208 22, 207.

9 26 R. Leipus and J. Šiaulys By the assumption of the theorem and 4.2wehavethatF Xn x =of Xη x for all n<κ. Thus, F Xκ x F Xκ x F Xη x F Xη x n<κ F Xn x F Xη x Pn η<κ Eϕη ηκ Eϕη κηk = Eϕη ηκ Eϕη κηk for every K > κ. Letting K, we get F Xκ x F Xκ x, proving that F Xκ L. Next, we show that F Xκ L F Xη L.Asin4.7, write F Xη x s x +s 2 x. Then, for large x, F Xη x F Xη x s x F Xη x + s 2x F Xη x s x s 3 x + s 2x, s 4 x where s 3 x :=F X κ xpη = κ;see4.2and4.8. Here, s x s 3 x = F Xn x n<κ F Xκ x Pn η<κ Pη = κ =0 by condition 2.3 and the assumption F Xκ L,and s 2 x s 4 x Eϕη ηκ Eϕη κηk by 4.. Thus, Letting K,weget F Xη x F Xη x Eϕη ηκ Eϕη κηk. F Xη x F Xη x. Proof of Theorem 4. i If η has finite port, pη 0,,...,m, then F Sη x = m P n= max kn k X i >x Pη = n. i=

10 On the random max-closure for heavy-tailed random variables 27 Thus, by inequality 4.9 and Lemma 3, F Sη x F Sη x max nm n pη Pmax k kn i= X i >x Pmax k kn i= X =. i >x ii For any K> and x K 2,wehave F Sη x = nk + K<nx/K =: p x+p 2 x+p 3 x. Write now p 3 x Pη >x/k=:p 3 x and p 2 x K<nx/K n PS k >xpη = n = P K<η x K Pη >K + x/l<kx/k kk kk PS k >x+ =: p 2 x+p 22 x+p 23 x, + PS k >x+ n>x/k K<kx/L PS k >xp k η x K PS n >xpη = n K<kx/K where L>Kand x L 2. We also need the following lower bound: F Sη x p x+p 2 x PS n >xpη = n+ nk =: p x+p 2x. PS k >xp k η x K PS k >xp k η x K K<nx/L PS n >xpη = n Now we can write, using inequality 4.9, F Sη x F Sη x p x max, p 22x p x p 2 x + p 23x F Sη x + p 2x F Sη x + p 3 x F Sη x 4.2 if K>, L>K,andx L 2. By Lemma 3, F Sn L for each n. Hence, by 4.9, p x p x max nk F Sn x =. 4.3 F Sn x Lith. Math. J., 572:208 22, 207.

11 28 R. Leipus and J. Šiaulys Next, we consider the bound for p 22 x /p 2 x. Letɛ be an arbitrary positive number, and let w.l.o.g. K>and L>beintegers. By Lemma 3 the d.f. F SK x is long tailed. Hence, F SK x F SK x +ɛ 4.4 for x M/2, wherem = MK is sufficiently large. Also, condition ii. of the theorem implies F Xk x F Xk x +ɛ 4.5 for all x L/2, sufficiently large L>M,andallk. By Lemma 4, F SK+ x F SK+ x max implying, due to estimates 4.4and4.5, yl/2 F XK+ y F XK+ y, yx L/2+ F SK y F SK y F SK+ x +ɛ. 4.6 xl F SK+ x Next, apply Lemma 4 again for the sum S K+2 = S K+ + X K+2 with v = x/2+/2: F SK+2 x F SK+2 x max yx/2+/2 If x 2L +, thenx/2+/2 L, sothat F SK+ y F SK+ y, yx/2+/2, F XK+2 y F XK+2 y F SK+2 x +ɛ 4.7 x2l + F SK+2 x due to estimates 4.5and4.6. Applying Lemma 4 once again with v =2x/3+/3, weget F SK+3 x F SK+3 x max y2x/3+/3 F SK+2 y F SK+2 y, yx/3+2/3 F XK+3 y F XK+3 y If x 3L +, then2x/3+/3 2L + and x/3+2/3 L. Hence, the last inequality, together with estimates 4.5 and4.7, implies Continuing, we obtain that for each k. F SK+3 x +ɛ. x3l + F SK+3 x F SK+k x +ɛ 4.8 xkl + F SK+k x..

12 On the random max-closure for heavy-tailed random variables 29 Since x /L L>Kfor x L 2, the set of indices k in K<kx /L is nonempty, and we have by 4.8and4.9that p 22 x K<kx /L p 2 x + ɛ PS k >xpη k K<kx /L PS k >xpη = k Pη k + ɛ 4.9 k>k Pη = k for x L 2 +, L>maxK, M. Next, we estimate p 23 x /F Sη x. Write p 23 x F Sη x k>x /L Pη k Pη =F X x Thus, condition ii.3 and the definition of class L imply Pη = Eη ηx /L. F X x Similarly, p 23 x = F Sη x p 3 x Pη >x /K F Sη x Pη =F X x K Eη η>x /K Pη = x F X x = Finally, for the term p 2 x /F Sη x, bylemma3 we have p 2 x F Sη x Pη >K = Pη >K min jk Pη = j max kk Relations 4.2, 4.3, 4.9, 4.20, 4.2, and 4.22 imply that F Sη x F Sη x kk F S k x kk F S k xpη = k F Sk x F Sk x Pη >K min jk Pη = j max, + ɛ k>k Pη k + Pη = k Pη >K min jk Pη = j for all ɛ>0 and all K>. Hence, letting K tend to infinity, by condition ii.2 of the theorem we get F Sη x F Sη x +ɛ for arbitrary ɛ>0. This proves that F Sη L. Lith. Math. J., 572:208 22, 207.

13 220 R. Leipus and J. Šiaulys 5 Examples Example. Let X,X 2,... be independent r.v.s such that X k is exponentially distributed for odd k,thatis, F Xk x =e x, x 0, k, 3, 5,..., and X 2,X 4,... are distributed according to the Peter and Paul law, that is, F Xk x = l: 2 l >x 2 l =2 log x/ log 2, x, k 2, 4,... Assume that η is a counting r.v., independent of X,X 2,..., with probabilities Pη = n =, n 0,,..., ζ5 n + 5 where ζs denotes the Riemann zeta function. It is easy to see that F X2 D although F X2 / L ;see[2], J + F X2 =, n2 n n F X k x/f X2 x <, andeη 3 <. Theorems and 2 imply that F Xη D and F Sη D. Example 2. Let X,X 2,... be independent r.v.s such that F Xk x = k e x, x 0, k, 2,..., and let η be a Poisson r.v. with parameter λ>0, independent of X,X 2,... ThenwehaveF Xk k and: L for Eϕη η E + log η η <, n F Xk x F X x = n k ϕn F Xk x =e x x k F Xk x Pη >n min kn Pη = k = condition 2.2 of Theorem 3i, assumption ii. of Theorem 4, k>n λk /k! λ n λe λ /n! n + 0 assumption ii.2 of Theorem 4, n Pη n Pη = n =+ λ n + + λ 2 n +n +2 + assumption ii.2 of Theorem 4, n Eη η>δx e δx Eηe η = o e x for all δ>0 assumption ii.3 of Theorem 4. Theorems 3 and 4 imply that F Xη L, F Sη L. Acknowledgment. We would like to thank the anonymous referees for their extremely valuable comments on the previous version of the manuscript.

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