Randomly weighted sums of subexponential random variables with application to capital allocation

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1 xtremes 204 7: DOI 0.007/s z Randomly weighted sums of subexponential random variables with application to capital allocation Qihe Tang Zhongyi Yuan Received: 2 October 203 / Revised: 5 April 204 / Accepted: 9 April 204 / Published online: 4 May 204 Springer Science+Business Media New York 204 Abstract We are interested in the tail behavior of the randomly weighted sum n θ i X i, in which the primary random variables X,...,X n are real valued, independent and subexponentially distributed, while the random weights θ,..., θ n are nonnegative and arbitrarily dependent, but independent of X,..., X n. For various important cases, we prove that the tail probability of n θ i X i is asymptotically equivalent to the sum of the tail probabilities of θ X,...,θ n X n, which complies with the principle of a single big jump. An application to capital allocation is proposed. Keywords Asymptotics Capital allocation Matuszewska indices Randomly weighted sum Subexponentiality AMS 2000 Subject Classifications Primary 6220; Secondary 60G70 Introduction Throughout the paper, all random variables are defined on the probability space,f, P unless otherwise stated. Let X,..., X n be n real-valued independent Q. Tang Department of Statistics and Actuarial Science, University of Iowa, 24 Schaeffer Hall, Iowa, IA 52242, USA qihe-tang@uiowa.edu Z. Yuan Department of Risk Management, Pennsylvania State University, 362 Business Building, University Park, PA 6802, USA ZhongyiYuan@psu.edu

2 468 Q. Tang, Z. Yuan random variables, called primary random variables, and let θ,..., θ n be n nonnegative random variables, called random weights, independent of the primary random variables. The target of this study is the randomly weighted sum S θ n = θ i X i. It is a central theme crossing various applied areas of probability and statistics to model the dependence structure of multiple random variables of interest, say, Z,..., Z n. This has become especially imperative in insurance, finance and risk management due to the increasing complexity of insurance and financial products. In lieu of the prevailing copula approach, we propose the stochastic representation Z i = θ i X i, i =,...,n, 2 with the primary random variables X,..., X n and the random weights θ,..., θ n described above. Our study will proceed in a general setting in which the random weights θ,..., θ n are arbitrarily dependent and not necessarily bounded. In the mechanism through 2, we intend to use the primary random variables X,..., X n to depict the magnitude of Z,..., Z n while using the random weights θ,..., θ n to capture the dependence. The independence between the two sequences {X,...,X n } and {θ,...,θ n } separates the magnitude and dependence, bringing us great convenience in both modeling and computation. In addition to an application to capital allocation to be given in Section 4, we list here three examples in insurance, finance and risk management in which a randomly weighted sum appears naturally as a key quantity of interest. In the first example, consider a portfolio consisting of n obligors that are subject to possible default over a given period. Denote by X i the loss given default of obligor i, equal to the percentage loss given default multiplied by the exposure, and denote by θ i the default indicator for obligor i, which is a Bernoulli random variable with θ i = corresponding to the default of obligor i. Then the randomly weighted sum represents the aggregate amount of losses. An important feature of credit risk is that the default indicators are strongly dependent due to both common macroeconomic factors and credit contagion. For detailed discussions, we refer the reader to McNeil et al. 2005, Das et al. 2007, and Hatchett and Kühn 2009, among many others. In the second example, consider an insurer who makes both risk-free and risky investments. Over each period i, the net loss is denotedby a real-valuedrandomvariable X i, equal to total claims minus total premiums, and the overall accumulation factor is denoted by a positive random variable A i. With θ i = i j= A j, the randomly weighted sum then represents the stochastic present value of the net losses

3 Randomly weighted sums of subexponential random variables 469 over the first n periods. See, for example, Nyrhinen 200 and Tang and Tsitsiashvili 2003a for related discussions. Obviously, the random variables θ,...,θ n should be strongly dependent. The third example gives another interpretation of in terms of an investment portfolio consisting of n risky assets over one period. ach asset i incurs a potential loss X i at the terminal time while the corresponding stochastic discount factor over the period is θ i. Then the randomly weighted sum represents the total amount of discounted losses potentially incurred from the investment portfolio. See, for example, Björk 2009 for discussions on stochastic discount factors. The stochastic discount factors θ,..., θ n should be dependent as they result from financial evolvements over the same time period. We study the tail behavior of the randomly weighted sum under the assumptions that the primary random variables are heavy tailed and that the random weights are dependent. These assumptions are highly relevant in the motivating examples above. Since many risk measures, such as Value at Risk VaR and Conditional Tail xpectation CT, focus on the tail area of a risk variable, the study of the tail behavior of randomly weighted sums has an immediate application to the computation of such risk measures. Recently, randomly weighted sums have been an attractive research topic in the literature of applied probability. See Shen et al. 2009, Zhang et al. 2009, Gao and Wang 200, Chen et al. 20, Yi et al. 20, Fougères and Mercadier 202, Hazra and Maulik 202 and Olvera-Cravioto 202, to name a few very recent ones. See also Hashorva et al. 200, who interpreted the stochastic representation 2 in terms of random contraction. Tang and Tsitsiashvili 2003b proved that, if the primary random variables X,..., X n are independent and identically distributed by a subexponential distribution, and the random weights θ,...,θ n take values in the interval a,b for some constants a and b,0<a b<,then n P Si θ >x P n Sn θ >x P θ i X i >x P θ i X i >x. 3 This suggests that the heavy tails of the primary random variables dissolve the dependence of the two-sided bounded random weights. This complies with the principle of a single big jump in the presence of random weights. Also, it can facilitate the computation of the tail probabilities, especially when there is tangled dependence among θ,...,θ n. However, the two-sided bound restriction on the random weights severely limits the merit of the result. Our study is a revisit to the work of Tang and Tsitsiashvili 2003b. We devote ourselves to relaxing the two-sided bound restriction. The lower-bound restriction is removed in Theorem while the upper-bound restriction is further removed in Theorems 2 and 3 under some additional conditions. A feature of our work is that the random weights are allowed to be arbitrarily dependent on each other.

4 470 Q. Tang, Z. Yuan The rest of the paper consists of five sections. In Section 2 we prepare some preliminaries on heavy-tailed distributions, in Section 3 we present our main results, in Section 4 we propose an application to capital allocation, and in Sections 5 and 6 we show the proofs. 2 Preliminaries 2. Heavy-tailed distributions Throughout this paper, all limit relationships are according to x unless otherwise stated. For two positive functions f and g, write fx gx or gx fxif lim sup fx/gx, write fx gx if lim fx/gx =, and write fx gx if f and g are weakly equivalent, that is, 0 < lim inf fx/gx lim sup fx/gx <. For real numbers x, x,..., x n, write x + = x 0, x = x 0 and x n = x x n. We shall assume that the primary random variables X,..., X n are heavy tailed, that is, their moment generating functions are infinite on 0,. One of the most important classes of heavy-tailed distribution functions is the subexponential distribution class. A distribution function F on R + =0, is said to be subexponential, written as F S, if it has an ultimate tail that is, Fx > 0 for all x 0 and F 2 x 2Fx, where F 2 denotes the two-fold convolution of F. More generally, a distribution function F on R is still said to be subexponential if the distribution function F + x = Fx x 0 is subexponential, where is the indicator function of a set. The class S contains a lot of important distributions such as Pareto, lognormal and heavytailed Weibull distributions. The reader is referred to the monographs mbrechts et al. 997, Asmussen and Albrecher 200 and Foss et al. 20 for reviews of the class S. Among many distribution classes that are closely related to the class S, we list a few here for our reference later. A distribution function F on R is said to be long tailed, written as F L, if its ultimate right tail satisfies Fx + y Fx for all y R. 4 Automatically, relation 4 holds uniformly on every compact set of y. Hence, it is easy to see that there is some positive function l, with lx x/2 and lx, such that relation 4 holds uniformly for lx y lx. Itis well known that every subexponential distribution is long tailed; see Lemma 2 of Chistyakov 964 or Lemma.3.5a of mbrechts et al By Theorem 2.a of Klüppelberg 988, a long-tailed distribution with a right tail weakly equivalent to a subexponential tail is subexponential. A distribution function F on R is said to be dominatedly-varying tailed, written as F D, if its right tail satisfies Fxy = O Fx for all 0 <y<.the

5 Randomly weighted sums of subexponential random variables 47 intersection L D forms a useful subclass of S; see, for example, Theorem of Goldie 978 or Proposition.4.4a of mbrechts et al In particular, it covers the famous class R of distributions with a regularly-varying tail. By definition, for a distribution function F on R, we write F R α for some 0 <α< if its right tail is regularly varying with index α,thatis,fxy y α Fxfor all y>0. In summary, R L D S L. In this paper, we need two more distribution classes. A distribution function F on R is said to belong to the class A if it is subexponential and its right tail satisfies Fxy lim sup < forsome y>. 5 x Fx Note that 5 is really a mild condition fulfilled by almost all useful distributions with an ultimate right tail. From this point of view, the class A almost takes up the entire class S. Moreover, a distribution function F on R is said to be rapidly-varying tailed, written as F R,ifFxy = o Fx for all y>. Note that R is a very broad class containing both heavy-tailed and light-tailed distributions. 2.2 Matuszewska indices The Matuszewska indices can provide useful information on the tail behavior of a distribution function. For a distribution function F with an ultimate right tail, define { } { M F = inf log F y : y> and M F = sup log F } y : y>, log y log y where F y = lim inf Fxy/Fxand F y = lim sup Fxy/Fx.FromCorollary 2..6 and Theorem 2..5 of Bingham et al. 987, we see that M F and M F are the upper and lower Matuszewska indices of the function f = /F, respectively. Without any confusion we simply call them the upper and lower Matuszewska indices of F. Let us recollectsome simple and usefulresults related to the Matuszewska indices. Clearly, two distributions with weakly equivalent tails have the same Matuszewska indices. It is easy to see that F D if and only if 0 M F <, or, equivalently, the function F has bounded decrease. Note also that inequality 5 holds if and only if 0 <M F, or, equivalently, the function F has positive decrease. Thus, F A can be characterized as F S with a tail possessing positive decrease. See page 7 of Bingham et al. 987 for the definitions of bounded decrease and positive decrease. For a distribution function F, with some simple adjustments on Proposition 2.2. of Bingham et al. 987, we see the following: if 0 M F <, then for every β > M F, there are some positive constants C and x such that the inequality Fx Fxy C y β 6

6 472 Q. Tang, Z. Yuan holds for all xy x x ; while if 0 <M F, then for every 0 <γ < M F, there are some positive constants C 2 and x 2 such that the inequality holds for all xy x x 2. From 6and7, it is easy to see that the relations Fxy Fx C 2y γ 7 x β = o Fx and Fx = o x γ 8 hold for every β>m F and γ < M F. Therefore, for a random variable X distributed by F,ifM F > then X + <, while if X + < then M F. 3 Main Results Recall the randomly weighted sum, in which the primary random variables X,...,X n are real valued, independent and distributed by F,...,F n, respectively, while the random weights θ,..., θ n are nonnegative, not degenerate at 0, and arbitrarily dependent on each other, but independent of the primary random variables. Here comes our first main result, which extends Theorem 3. of Tang and Tsitsiashvili 2003b by squarely removing the lower-bound restriction on the random weights: Theorem If F i L and F i x Fx for some F S and all i =,...,n, and if θ,...,θ n are bounded above, then the relations in 3 hold. Then we aim to further remove the upper-bound restriction on the random weights. In doing so, we need to confine the primary distributions to the class A, whichis slightly smaller than the class S, as mentioned before. Theorem 2 If F i L and F i x Fxfor some F A and all i =,...,n, and if the relation P θ i >ux = op θ i X i >x, u > 0, 9 holds for all i =,...,n, then the relations in 3 hold. Lemma3.2ofTang2006 shows that 9 is equivalent to the existence of a positive auxiliary function a, with ax and ax = ox, such that P θ i >ax = op θ i X i >x. 0 As shown in Corollary 2. of Tang 2006, either one of the following is sufficient for 9 to hold: P θ i >xy = o F i x for some y>0; P θ i >xy = op θ i >x for some y>;

7 Randomly weighted sums of subexponential random variables 473 X + βi 0 <M G i and i = for some 0 <β i <M G i,where G i is the distribution function of θ i. In the following theorem, we confine the primary distributions to the class L D but do not assume weak equivalence for their tails. It slightly extends Theorem 2. of Wang and Tang Theorem 3 If F i L D and θ β i i < for some β i >M F i and all i =,...,n, then the relations in 3 hold. We postpone the proofs of Theorems 3 to Section 5. 4 Application to Capital Allocation Consider an investor who invests in n lines of business. ach line i generates a potential net loss variable Z i in loss-profit form. These loss variables must depend on each other since the n lines operate in similar macroeconomic environments. We model them by 2, in which, as described before, the primary random variables X,..., X n are real valued, independent and distributed by F,..., F n, respectively, while the random weights θ,..., θ n are nonnegative, not degenerate at 0, and arbitrarily dependent on each other, but independent of the primary random variables. Then the randomly weighted sum Sn θ = θ i X i represents the total loss. For some regulated investors, such as banks or insurance companies, a risk capital is held as a cushion to protect them from large losses. Our interest is in allocating the risk capital to the individual lines. We refer the reader to Bauer and Zanjani 204 for an overview of various allocation principles, many of which boil down to uler s principle, also referred to as gradient allocation principle. uler s principle allocates risk capital according to the risk contribution of each line and is known to provide right signals for performance measurement; see also Tasche 999 for related discussions. Moreover, uler s principle is coherent if the underlying risk measure is coherent; see Denault 200 for an introduction of coherent allocation principles. For more details and economic justifications of uler s principle, we refer the reader to Myers and Read 200, McNeil et al and Dhaene et al. 202, among others. See also Asimit et al. 20 for an asymptotic analysis of capital allocation in the presence of extreme risks, to which our study in this section is similar in spirit. Assume that each random variable in has a finite mean and let CT be the underlying risk measure. According to uler s principle, the amount of capital allocated to line i is AC i = θ i X i θ i X i Sn θ S θ >x n >x = P Sn θ, i =,...,n, >x

8 474 Q. Tang, Z. Yuan where x is the VaR of Sn θ,thatis, { x = VaR q S θ n = inf y R : P S θ n y q }, 0 <q<. See Section of McNeil et al for this assertion. Note that AC i given by could be nonpositive, namely, no capital or even a negative amount of capital is allocated to line i in this case, meaning that such a line should be rewarded for risk capital. See rel et al. 204 for related discussions. Typically, the expression given by is difficult to evaluate. We aim at an asymptotic formula as q. Under the corresponding conditions of each of Theorems 3, the asymptotics for the denominator in has been given by 3, which consequently leads to } VaR q S θ n inf {x R : P θ i X i >x q, q. Thus, only asymptotics for the numerator in needs to be derived. We show the results for i = only. Theorem 4 In addition to the conditions of Theorem, Theorem 2 or Theorem 3 assume that P θ i X i >x = OP θ X >x, i = 2,...,n. 2 Then we have θ X S θ n >x θ X θ X >x. 3 Within the scope of Theorem or Theorem 2, a sufficient condition for 2 is that P θ i >t = O P θ >t as t ˆt for all i = 2,...,n,where0< ˆt is the essential upper bound of θ ; while within the scope of Theorem 3, a sufficient condition for 2isthatF 2 x + +F n x = O F x. In the next theorem for the L D case, we do not require condition 2 but slightly strengthen the moment conditions on the random weights. Theorem 5 If F i L D and θ β + θ θ β i i < for some β i >M F i and all i = 2,...,n,then and θ X + S θn >x θ X S θn >x θ X + θ i X i >x θ X θ i X i >x, where the second relation is meant for the case that X has a nontrivial negative part X.

9 Randomly weighted sums of subexponential random variables 475 The two asymptotic relations in Theorem 5 readily give that θ X S θ n >x n θ X θi X i >x under a mild technical requirement that lim inf x n θ X + θ i X i >x n θ X θ i X i >x > or lim sup x n θ X + θ i X i >x n θ X θ i X i >x <. We make the right-hand side of 3 explicit in the next corollary by imposing some more conditions. In order to state the corollary below, we introduce two nonnegative random variables X and θ, which are independent of each other and independent of all other sources of randomness. Let X be distributed by F and denote by ˆt the essential upper bound of θ. We show some cases below in which the asymptotics for the capital allocation becomes transparent: Corollary Assume that, for each i =,...,n, F i x c i Fxfor some constant c i > 0. a For F S R and 0 < ˆt <, assume that lim t ˆt P θ i >t /P θ >t = d i for some d > 0, d 2 0,...,d n 0. Then, as q, AC c d n VaR q θx with q = q c i d n. 4 i c i d i b c For F S R and ˆt =, assume that lim t ˆt P θ i >t /P θ >t = d i for some d > 0, d 2 0,..., d n 0 and that relation 9 holds for all u>0 and all i =,...,n. Then relation 4 still holds. For F R α with α>, assume that θ β i < for some β>αand all i =,...,n.then AC α c θ α α n c i θi α /α VaR q X, q. We postpone the proofs of Theorems 4 5 and Corollary to Section 6. 5 Proofs of Theorems 3 For each of Theorems 3, the last relation in 3 can be easily proven by verifying j =k n P θ j X j >x,θ k X k >x = o P θ i X i >x. 5 For Theorem, relation 5 is trivial because the random weights are bounded. For Theorem 2, relation 5 can be verified by conditioning on θ j and θ k for each term on the left-hand side and utilizing relation 0. For Theorem 3, relation 5 follows from Lemma 7 below.

10 476 Q. Tang, Z. Yuan and The rest of 3 amounts to the conjunction of P S θ n >x P θ i X i >x 6 P θ i X i + >x P θ i X i >x. 7 For this reason, we shall focus on 6and7 only in the proofs of Theorems Proof of Theorem Following the proof of Proposition 5. of Tang and Tsitsiashvili 2003b with some obvious changes, we obtain Lemma Let X,..., X n be n real-valued independent random variables, each distributed by F i satisfying F i L and F i x Fx for some F S and all i =,...,n. Then for every fixed 0 < a b <, it holds uniformly for all c,...,c n a,b n that P c i X i >x P c i X i >x. 8 The following lemma can be easily verified by conditioning on θ and utilizing relation 0; see also Theorem 2.2iii of Cline and Samorodnitsky 994: Lemma 2 Let X and θ be two nonnegative independent random variables. If X is long tailed, θ is not degenerate at 0, and P θ >ux = op θx > x for all u>0, then the product θx is long tailed. Proof of Theorem Without loss of generality, we assume that the random weights θ,...,θ n are bounded above by. To prove 6, first assume that the random variables X,...,X n are nonnegative. For this case, relation 6 follows from the inequality S θ n n θ i X i and relation 5. Now consider the general case where X,..., X n may be negative. For an arbitrary subset I {,...,n}, write I c ={,...,n}\i and I X = { ω : X i 0fori I and X j < 0forj I c}. Since each random weight is bounded above by, it follows that P Sn θ >x P θ i X i + X j >x, I X, 9 i I j I c =I {,...,n} where a sum over an empty set is equal to 0 by convention. Conditioning on X j for j I c and applying relation 6 for the nonnegative case, we see

11 Randomly weighted sums of subexponential random variables 477 that each probability on the right-hand side of 9 is asymptotically not smaller than P θ i X i + X j >x, I X P θ i X i >x, I X, i I j I c i I where we further applied the dominated convergence theorem and the fact that θ i X i is long tailed due to Lemma 2. Substituting this into 9 and interchanging the order of the summations yield relation 6. Then we turn to establishing 7. First we assume that the random weights are positive. Let I and I c be as before and write ε I θ = { ω : θ i >εfor i I and θ j ε for j I c}, 0 <ε<. Clearly, P θ i X i + >x I {,...,n} P θ i X i + + εx j + >x, ε I θ. 20 i I j I c By Lemma, each term on the right-hand side of 20 is asymptotically equivalent to P θ i X i >x, ε I θ + P εx j >x P ε I θ j I c i I = i I P θ i X i >x, ε I θ + j I c P εx j >x,θ j >ε P ε I θ P θ j >ε. Substituting this into 20 and interchanging the order of the summations, we obtain P θ i X i + >x I:i I {,...,n} P θ i X i >x, ε I θ + j= I:j/ I {,...,n} = P θ i X i >x,θ i >ε + P θ j X j >x P θ j ε P θ j= j >ε P θ j ε + max j n P θ j >ε P θ i X i >x. Since each θ j is strictly positive, letting ε 0 yields 7. P θ j X j >x P ε I θ P θ j >ε

12 478 Q. Tang, Z. Yuan Finally, consider the general case where the random weights may take value 0 with a positive probability. Let I and I c be as before and write 0 I θ = { ω : θ i > 0fori I and θ j = 0forj I c}. Then, by what we have just proven for 7, P θ i X i + >x = = This ends the proof of Theorem. 5.2 Proof of Theorem 2 φ =I {,...,n} φ =I {,...,n} i I P θ i X i + >x, 0 I θ i I P θ i X i >x, 0 I θ P θ i X i >x. 2 The following is an extension of Lemma 3. of Tang 2006: Lemma 3 In addition to the conditions of Lemma, assume that M F > 0. Then it holds uniformly for all c,...,c n 0, n that P c i X i >x P c i X i >x. Proof If we rewrite P n c i X i >x = P n c i X i > x with c i = c i /c n and x = x/c n, then each c i lies in 0,, at least one of them equals, and x. Thus, without loss of generality, we may assume c =. For every 0 <ε<andi {2,...,n}, write I c ={2,...,n}\I and A I = { c 2,...,c n : 0 <c i ε for i I and ε<c j forj I c}. By Lemma and inequality 7, uniformly over each A I it holds for arbitrarily fixed 0 <γ <M F and some C>0that P n c i X i >x P X + n P c i X i >x + i I εx+ i + j I c c j X j + >x P X >x + j I c P c j X j >x P X >x + i I P εx i >x + j I c P c j X j >x P X >x + j I c P c j X j >x + P εx i >x P X i I >x + Cε γ n.

13 Randomly weighted sums of subexponential random variables 479 Since {A I : I {2,...,n}} forms a finite partition of 0, n and ε can be arbitrarily close to 0, the result follows. The following lemma complements Lemma 3 with the opposite inequality: Lemma 4 Let X,..., X n be n nonnegative independent random variables, each with an ultimate right tail. Then it holds uniformly for all c,...,c n 0, n that P c i X i >x P c i X i >x. Proof Note that P n c i X i >x P n c i X i >x. A simple application of Bonferroni s inequality completes the proof. A combination of Lemmas 3 and 4 extends Lemma as follows: Lemma 5 Let X,..., X n be n nonnegative independent random variables, each distributed by F i satisfying F i L and F i x Fx for some F A and all i =,...,n. Then relation 8 holds uniformly for all c,...,c n 0, n. The following lemma plays a crucial role in proving relation 6 for Theorem 2: Lemma 6 Let X, {Y,...,Y n } and {θ 0,θ,...,θ n } be three independent groups of nonnegative random variables. If X is distributed by F L with M F > 0, and P θ i >ux = op θ i X>x for all u>0 and i = 0,,...,n,then P θ 0 X θ i Y i >x = P θ 0 X>x o P θ i X>x. Proof Since the inequality P θ 0 X n θ i Y i >x P θ 0 X>x is trivial, we only need to prove the opposite inequality. As remarked after Theorem 2, there is some positive auxiliary function a, with ax and ax = ox, such that Pθ i > ax = opθ i X>xholds for all i = 0,,...,n.Letθ n = n θ i. For arbitrarily fixed 0 <ε<, we derive P θ 0 X θ i Y i >x P θ 0 X θ i Y i >x,εθ n <θ 0 ax P θ 0 X θ 0 ε Y i >x,εθ n <θ 0 ax P θ 0 X>x,εθ n <θ 0 ax P θ 0 X>x P θ 0 X>x,εθ n θ 0 Pθ0 > ax, 22

14 480 Q. Tang, Z. Yuan where in the third step we conditioned on θ 0 and applied the dominated convergence theorem and F L. It holds for every 0 <γ <M F, somec>0 and all large x that P θ 0 X>x,εθ n θ 0 P εθn X>x P εθ n X>x,θ n ax + P θ n >ax Cε γ P θ n X>x + P θ i > ax Cε γ P θ i X>x, where the third step is due to relation 7. Substituting this into 22, by the arbitrariness of ε we conclude the proof. Proof of Theorem 2 Temporarily assume that the random weights are positive. First we prove 6. For an arbitrary subset I {,...,n}, write I c = {,...,n}\i and I X = { ω : X i 0fori I, X j < 0forj I c}. Recall the equivalence between 9 and0. With the positive auxiliary function a specified in relation 0 for all i =,...,n, it holds that P Sn θ >x =I {,...,n} P i I θ i θ n X i + j I c θ j X j > x,θ n ax, I X. θ n θ n By conditioning on θ,...,θ n and X j for j I c and applying Lemma 4, we obtain P S θ n >x =I {,...,n} i I =I {,...,n} i I P θ i X i + θ j X j > x,θ n ax, I X θ n θ j I c n θ n P θ i X i + θ j X j >x, I X P θ n >ax. j I c

15 Randomly weighted sums of subexponential random variables 48 Note that P θ j X i >x P θ i X i >x holds for all i, j n. Applying Lemma 6 and interchanging the order of the summations yield P S θ n >x = =I {,...,n} i I P θ i >ax P θ i X i >x, I X o P θ j X i >x j I c P θ i X i >x o o P θ i X i >x P θ i X i >x. Next we prove 7. Clearly, P =I {,...,n} i I,j I c θj X i >x P θ i X i + θ i >x P X i + > x,θ n ax + P θ n >ax θ n θ n P θi θ n X i > P θ i X i >x, x θ n,θ n ax + P θ i >ax where in the second step we applied Lemma 5. So far we have proven Theorem 2 for the case with positive random weights. The extension to the case with nonnegative random weights can be done by copying Proof of Theorem 3 The following lemma is a slight extension of Lemma 6.2 of Tang and Yuan 202 and is at the core of the proof of Theorem 3: Lemma 7 Let X be a random variable with a dominatedly-varying right tail and upper Matuszewska index M,letθ be a nonnegative random variable with θ β < for some β > M,let{ t,t T } be a set of random events satisfying lim t t0 P t = 0 for some t 0 in the closure of the index set T, and let

16 482 Q. Tang, Z. Yuan {θ,{ t,t T }} be independent of X. Then P θx > x, t P θx > x, t lim lim sup = lim lim sup = 0. t t 0 x P θx > x t t0 x P X >x Proof Since P θx > x P X >x by Theorem 3.3iv of Cline and Samorodnitsky 994, we only need to prove the second relation. Choose some c such that M <cβ<βand do the split P θx > x, t = P θx > x, t,θ x c + P θx > x, t,θ >x c. By inequality 6, there is some constant C>0 such that, for all large x, P θx > x, t,θ x c CP X >x θ β t. Moreover, by Markov s inequality and the second relation in 8, P θx > x, t,θ >x c P θ>x c x cβ θ β = op X >x. By these upper bounds we conclude the proof. Proof of Theorem 3 We first prove 6. Since each θ i X i is long tailed by Lemma 2, we can choose some positive function l, with lx and lx x/2, such that the relation P θ i X i >x+ y P θ i X i >x holds uniformly for lx y lx and i =,...,n. By Bonferroni s inequality, we have P Sn θ >x n P Sn θ >x, θ i X i >x+ lx P Sn θ >x,θ ix i >x+ lx j<k n = + o + o P θ j X j >x+ lx, θ k X k >x+ lx P θ i X i >x+ lx P θ i X i >x P θ i X i >x, P Sn θ x,θ ix i >x+ lx P θ i X i >x+ lx, where in the third and last steps we applied Lemma 7. j=,j =i θ j X j < lx

17 Randomly weighted sums of subexponential random variables 483 To show relation 7, we observe that P θ i X i + >x n P θ i X i >x lx n n + P θ i X i + >x, θ j X j x lx, θ k X k > x n j= k= P θ i X i >x lx + P θ k X k > x n, θ i X i + >lx k=,i =k P θ i X i >x, where in the last step we applied Lemma 7 again. The proof is complete. 6 Proofs of Theorems 4 5 and Corollary In the proofs of Theorems 4 5, we define a new probability measure Q on,f by dq dp = θ X + θ X +, 23 where, and throughout this section, the symbol without a superscript means the expectation still under P. Under Q, it is easy to verify the following facts: the two sequences {X,...,X n } and {θ,...,θ n } are still independent; the primary random variables X,..., X n are still independent with the distribution functions of X 2,..., X n remaining intact; the tail probabilities of X and θ X are given by Q X >x = xf x + x F ydy, x > 0, X and Q θ X >x = xpθ X >y+ x Pθ X >ydy, x > 0, θ X respectively, implying that the distribution functions of X and θ X are both long tailed. In the proofs below, we shall often apply these facts tacitly.

18 484 Q. Tang, Z. Yuan 6. Proof of Theorem 4 Clearly, it suffices to prove that θ X S θ n >x θ X θ X >x 24 and that θ X + n θi X +i >x θ X θ X >x. 25 To prove relation 24, note that θ X S θ n >x = θ X + S θn >x θ X S θn >x θ X + θ X n θi X i >x θ X n θ i X i >x = θ X + Q θ X θ i Xi >x X θ n θ i X i >x, while to prove relation 25, note that, for x>0, θ X + n θi X +i >x = θ X θ X >x + θ X + θ X + x, n θ i X +i >x. Thus, it suffices to prove the following: Q θ X θ i Xi >x Q θ X >x ; 26 θ n θ i X i >x = o θ X θ X >x ; 27 θ X + θ X + x, n θi X +i >x = o θ X θ X >x. 28 Assuming that relation 2 is satisfied, we shall prove relations under the conditions of Theorem, Theorem 2 or Theorem 3. a Assume that the conditions of Theorem hold, and assume without loss of generality that the random weights θ,...,θ n are bounded above by. Relations 26 and 27 hold straightforwardly since Q θ X θ i Xi >x Q θ X Xi >x Q θ X >x and θ n θ i X i >x P θ i X i >x = o θ X θ X >x,

19 Randomly weighted sums of subexponential random variables 485 where in the last step we applied Theorem, relation 2, and the fact that xp θ X >x θ X θ X >x. To prove relation 28, note that, for x>0, θ X + θ X + x, n θi X +i >x xp θ i X i + x<θ >x + xp X x, θ i X i + >x. 29 By Theorem and relation 2, we have P θ i X i + >x P θ i X i >x = OP θ X >x. 30 Moreover, it holds that P x<θ X x, θ i X i + >x P θ i X i + >x P θ X >x P X x P θ i X i + >x = op θ X >x, 3 where we applied Theorem again in the last step. A combination of relations 29 3 yields relation 28. b Assume that the conditions of Theorem 2 hold. First, to prove relation 26, observe that, for every u>0, Q θ >ux = = u 0 = o P θ > ux s ds 0 0 P θ >ux s ds = oq θ X >x. P θ X > x s ds Consequently, there is some positive auxiliary function ã, with ãx and ãx = ox, such that Q θ > ãx = oq θ X >x. 32 Similarly to the derivation of relation 22, we have, for θ n = n θ i and arbitrarily fixed 0 <ε<, Q θ X θ i Xi >x Q θ X >x Q θ X >x,εθ n θ Q θ > ãx. 33

20 486 Q. Tang, Z. Yuan By relation 2, there is some C>0 such that, for x>0, Q θ X >x,εθ n θ = θ X + θ X θ X >x, εθ n θ θ X + εθ n X θn X >x ε = θ X + P θ n X >x s ds 0 Cε θ X + P θ X >x s ds 0 = CεQ θ X >x. Substituting this and 32 into33 and noticing the arbitrariness of ε, we obtain 26. Next, to prove relation 27, we derive θ n θ i X i >x = θ θ ãx, n θ i X i >x ãxp θ i X i >x ãx + θ θ >ãx, n θ i X i >x + θ θ >ãx P θ i X i >x + θ Q θ > ãx = o θ X θ X >x, where we applied Theorem 2 in the third step and relation 32 in the last step. Finally, we prove relation 28. For every 0 <ε<, it holds for x>0that θ X + θ X x, n θi X +i >x εxp θ i X i + >x + xp εx < θ X x, θ i X i + >x. By Theorem 2 and relation 2, we have P θ i X i + >x P θ i X i >x = OP θ X >x.

21 Randomly weighted sums of subexponential random variables 487 With the auxiliary function a given in 0, we have P εx < θ X x, θ i X i + >x P θ i X i + >x P θ X >x P X ε x P θ ax, ax = op θ X >x, θ i X i + >x where in the last step we applied Theorem 2 twice. By the arbitrariness of ε, we conclude that θ X + θ X x, n θi X +i >x = oxp θ X >x = o θ X θ X >x. c Assume that the conditions of Theorem 3 hold. First, to prove relation 26, observe that there is some positive function l, with lx x/2 andlx, such that Q θ X >x+ lx Q θ X >x. Also observe that the distribution function of X under Q has an upper Matuszewska index not greater than M F 0, and that Q θ β Q θ X = θ i Xi >x Q θ X >x+ lx, θ β θ i Xi / θ <. Wehave lx = Q θ X >x+ lx Q θ X >x+ lx, Q θ X >x, θ i Xi >lx where in the last step we applied Lemma 7. Next, to prove relation 27, choose some c such that M F <cβ <β.for every ε>0, it holds for x>0that θ n θ i X i >x = θ θ x c, n θ i X i >x x c P θ i X i >x ox P θ i X i >x+ oxp θ X >x+ + θ θ >x c, n θ i X i >x + θ θ >x c θ β θ β /β P θ >x c β /β /β + x cβ,

22 488 Q. Tang, Z. Yuan where we applied Theorem 3 and Hölder s inequality in the third step and Markov s inequality in the last step. Since cβ + > M F and P θ X >x P X >x,wehavex cβ = oxp θ X >x. Therefore, relation 24 holds. Finally, to prove relation 28, let c and c 2 be arbitrarily fixed constants satisfying 0 <c 2 <c < andm F <c 2 β.wehave θ X + θ X x, n θi X +i >x x c P θ i X i + >x + xp x c <θ X x, θ i X i + >x. By Theorem 3 and relation 2, we have P θ i X i + >x Furthermore, it holds that P θ i X i >x = OP θ X >x. P x c <θ X x, θ i X i + >x P θ i X i + >x P θ X >x P X x c c 2 P θ x c 2, = op θ X >x, θ i X i + >x where in the last step we applied Theorem 3 again and the fact that P θ >x c 2 = opθ X >x. It follows that θ X + θ X x, n θi X +i >x = oxp θ X >x = o θ X θ X >x. The proof is complete. 6.2 Proof of Theorem 5 It is easy to verify that, under Q defined by 23, the distribution function of X still belongs to the class L D with an upper Matuszewska index not greater than M F 0,. Moreover, it holds for every i = 2,...,nthat θ β Q θ β θ θ β i = < and Q θ β i i i = <. 34 θ θ

23 Randomly weighted sums of subexponential random variables 489 Therefore, an application of Theorem 3 immediately gives θ X + S θn >x = θ X + Q S θ n >x θ X + n Q θ i X i >x = θ X + θ i X i >x. When X has a nontrivial negative part X, define another probability measure Q on,f by dq dp = θ X θ X. Note that under Q all distributional results of {X 2,...,X n ; θ,...,θ n } remain the same as under Q. Thus, by the moment conditions in 34 and Theorem 3, we have Q θ i X i >x Q θ i X i >x. 35 On the one hand, it follows from relation 35that θ X S θn >x θ X n θ i X i >x = θ X Q θ i X i >x θ X n Q θ i X i >x. 36 On the other hand, to derive an asymptotic lower bound for θ X S θn >x, observe that by Lemma 2 the distribution function of each product θ i X i under Q belongs to L D. Hence, there is some positive function l, with lx and lx x/2, such that Q θ i X i >x+ lx Q θ i X i >x, i = 2,...,n. 37 Moreover, θ X S θn >x θ X n θi Xi >x+lx,θ X lx = θ X n θ i X i >x+lx θ X n θi Xi >x+lx,θ X >lx.

24 490 Q. Tang, Z. Yuan By relations 35and37, θ X n θ i X i >x+lx = θ X Q θ i X i >x+ lx θ X n Q θ i X i >x. Also, note that θ X n θi Xi >x+lx,θ X >lx θ X θ i X i > n x,θ X >lx = θ X Q = o Q θ i X i >x, θ i X i > x n,θ X >lx where the last step is due to Lemma 7. We then have θ X S θn >x θ X n Q θ i X i >x, which together with relation 36gives θ X S θn >x θ X θ i X i >x = θ X θ i X i >x. This completes the proof. 6.3 Proof of Corollary a For each i =,...,n, by conditioning on θ i we obtain P θ i X i >x c i P θ i X>x. For every ε>0, there is some small 0 <δ<ˆt such that P θ i >t d i + ε P θ >t holds for all ˆt δ<t ˆt. SinceX is rapidly-varying tailed, we have P θ i X i >x P θx > x c P θ i X>x i P θx > x c P θ i X>x,θ i > ˆt δ i P θx > x,θ > ˆt δ c id i + ε, where the last step is obtained by conditioning on X in both the numerator and the denominator. The opposite asymptotic inequality can be established similarly. Thus, by the arbitrariness of ε we obtain P θ i X i >x lim x P θx > x = c id i, i =,...,n. 38

25 Randomly weighted sums of subexponential random variables 49 It follows from Theorem and relation 38that and, hence, that P S θ n >x P θx > x c i d i 39 x = VaR q S θ n VaR q θx, q. 40 Moreover, since the product θx is rapidly-varying tailed, by 7, for arbitrarily fixed γ>, there is some C>0 such that P θx > xy P θx > x Cy γ holds for all large x and all y. Therefore, by the dominated convergence theorem and relation 38, P θ X >xy θ X θ X >x = xp θ X >x + c d xp θx > x + P θ X >x dy P θx > xy P θx > x dy c d xp θx > x. 4 Starting from relation and applying Theorem 4a and relations 39 4, we obtain AC θ X θ X >x P θx > x n c d c i d n VaR q θx, q. i c i d i b The proof is similar to that of a and, hence, is omitted. c By Theorem 3 and Breiman s theorem see Breiman 965 and Cline and Samorodnitsky 994, we have P S θ n >x Fx c i θi α. 42 By Proposition 0.8v of Resnick 987, F /x is regularly varying with index /α. Therefore, as q, x = VaR q S θ n F q n c i θi α c i θ α /α i VaR q X. 43

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