PRODUCTS IN CONDITIONAL EXTREME VALUE MODEL

Transcription

1 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL RAJAT SUBHRA HAZRA AND KRISHANU MAULIK Abstract. The classical multiariate extreme alue theory tries to capture the extremal dependence between the components under a multiariate domain of attraction condition and it requires each of the components to be in the domain of attraction of a uniariate extreme alue distribution as well. The multiariate extreme alue MEV model has a rich theory but has some limitations as it fails to capture the dependence structure in presence of asymptotic independence. A different approach to MEV was gien by Heffernan and Tawn 2004, where they examined MEV distributions by conditioning on one of the components to be extreme. Here we assume one of the components to be in Frechét or Weibull domain of attraction and study the behaior of the product of the components under this conditional extreme alue model.. Introduction The classical multiariate extreme alue theory tries to capture the extremal dependence between the components under a multiariate domain of attraction condition and it requires each of the components to be in domain of attraction of a uniariate extreme alue distribution. The multiariate extreme alue theory has a rich theory but has some limitations as it fails to capture the dependence structure. The concept of tail dependence is an alternatie way of detecting this dependence. The concept was first proposed by Ledford and Tawn 996, 997 and then elaborated upon by Maulik and Resnick 2004, Resnick A different approach towards modeling multiariate extreme alue distributions was gien by Heffernan and Tawn 2004 by conditioning on one of the components to be extreme. Further properties of this conditional model were subsequently studied by Das and Resnick 20, Heffernan and Resnick One of the limitations of the asymptotic independence model is that it is too large a class to conclude anything interesting, for example, about product of two random ariables. In another approach, the concept was thus weakened by Maulik et al. 2002, where they assumed that, Y satisfy the following ague conergence: at, Y ν H on M + 0, 0,,. where M + 0, 0, denotes the space of nonnegatie Radon measures on 0, 0, and νx, = x α, for some α > 0 and H is a probability distribution supported on 0,. The tail behaior of the product Y under the assumption. and some further moment conditions was obtained by Maulik et al The conditional model can be iewed as an extension of the aboe model. Under the conditional model, the limit of the ague conergence need not be a product measure and it happens in the space M +, E γ with γ R, where E γ is the right closure of the set {x R : + γx > 0}. See Section 2 for details. In this article we mainly focus on the product behaior when the limiting measure in the conditional model is not of the product form. Products of random ariables and their domain of attraction are important theoretical issues which hae a lot of applications ranging from Internet traffic to insurance models. We study the product of two random ariables whose joint distribution satisfies the conditional extreme alue model. In particular we try to see the role of regular ariation in determining the behaior of the product. When γ > 0, then it is easy to describe the behaior of the product under certain conditions. Howeer, when γ < 0, some complications arise due to the presence of finite upper end point. We remark that in this paper we do not deal with the other important case of γ = Mathematics Subject Classification. Primary60G70; Secondary62G32. Key words and phrases. Regular ariation, domain of attraction, generalized extreme alue distribution, heay tails, asymptotic independence, conditional extreme alue model, product of random ariables.

2 2 R. S. HAZRA AND K. MAULIK In Section 2 we briefly describe the conditional extreme alue model and state some properties and nomenclature which we use throughout this article. In Section 3 we proide an oeriew of our results presented in the later sections. In Section 4 we reduce the conditional extreme alue model defined in Section 2 to simpler forms in special cases. In Section 5 we describe the behaior of the product of random ariables following conditional model under some appropriate conditions. In Section 6 we make some remarks on the assumptions used in the results. In the final Section 7 we present an example of a conditional model and also look at the product behaior. 2. Conditional extreme alue model In this section, we proide the notations used in this paper and the basic model. If S is a topological space with S being its σ-field, then for non-negatie Radon measures µ t, for t > 0, and µ on S, S, we say µ t conerges aguely to µ and denote it by µ t µ, if for all relatiely compact sets C, which are also µ-continuity sets of µ, that is, µ C = 0, we hae µ t C µc, as t. By M + S we shall mean the space of all Radon measures on S endowed with the topology of ague conergence. Definition 2.. A measurable function f : R + R + is called regularly arying at infinity with index α, we write f RV α, if for all t > 0, ftx lim x fx = tα. If α = 0, f is called slowly arying. Definition 2.2. We say that a random ariable, with distribution function F, has a regularly arying tail of index α, α 0, if the tail of its distribution function F := P > RV α. Definition 2.2 is equialent to the existence of a positie function a RV /α such that /at has a ague limit in M + 0,, where the limit is a nondegenerate Radon measure. The limiting measure necessarily takes alues cx α on set x,. Let E γ be the interal {x R : + γx > 0} and E γ be its right closure in the extended real line,. Thus, we hae /γ,, if γ > 0, /γ,, if γ > 0, E γ =,, if γ = 0, and E γ =,, if γ = 0,, /γ, if γ < 0,, /γ, if γ < 0. For any γ R, the generalized extreme alue distribution is denoted by G γ. It is supported on E γ and is be gien by, for x E γ, { exp + γx γ, for γ 0, G γ x = exp e x, for γ = 0. Definition 2.3. We say that a random ariable Y with distribution function F is in the domain of attraction of an extreme alue distribution G γ for some γ R written as DG γ if there exists a positie alued function a and a real alued function b such that as t, on E γ, { + γy γ, for γ 0, Y > aty + bt = tf aty + bt 2. e y, for γ = 0. When γ 0, the domain of attraction condition is related to regular ariation in the following way. If γ > 0, then F DG γ if and only if F RV /γ. If γ < 0, then as t, bt b < and F DG γ if and only if F b / RV /γ. Note that in this case b becomes the upper end point of the distribution function F. Definition 2.4 Conditional extreme alue model. The real alued random ector, Y satisfies conditional extreme alue model CEVM if The marginal distribution of Y is in the domain of attraction of an extreme alue distribution G γ, that is, there exists a positie alued function a and a real alued function b such that 2. holds on E γ.

3 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 3 2 There exists a positie alued function α and a real alued function β and a non-null Radon measure µ on Borel subsets of, E γ such that γ 2A µ on, E, as t, and βt αt, Y bt at 2B for each y E γ, µ, x y, is a nondegenerate distribution function in x. 3 The function Hx = µ, x 0, is a probability distribution. If, Y satisfy Conditions 3, then we say, Y CEV Mα, β; a, b; µ in, E γ. Note that H is a nondegenerate probability distribution function by Condition 2B. Also, Condition 2A is equialent to the conergence αtx + βt, Y > aty + bt µ, x y, for all y E γ and continuity points x, y of the measure µ. Note that, for, Y CEV Mα, β; a, b; µ in, E γ we hae, as t, βt P x αt Y > bt Hx, which motiates the name of the model. Occasionally, we shall also be interested in pairs of random ariables, Y which satisfy Conditions 2A and 2B, without any reference to Conditions and 3. We shall then say that, Y satisfies Conditions 2A and 2B with parameters α, β; a, b; µ on E, where α and β will denote the scaling and centering of, a and b will denote the scale and centering of Y and µ will denote the nondegenerate limiting distribution and E is the space on which the conergence takes place. In Definition 2.4, we hae E =, E γ. It was shown in Proposition of Heffernan and Resnick 2007 that there exists functions ψ, ψ 2 such that, αtx lim t αt = ψ βtx βt x and lim = ψ 2 x t αt and the aboe conergence holds uniformly on compact subsets of 0,. Then, necessarily, we must hae, for some ρ R, ψ x = x ρ, x > 0. Also from Theorem B.2. of de Haan and Ferreira 2006 it follows that either ψ 2 is 0 or, for some k R and x > 0, { k ρ ψ 2 x = xρ, when ρ 0, k log x, when ρ = 0. Note that, we hae α RV ρ. Definition 2.5. The pair of nonnegatie random ariables Z, Z 2 is said to be standard multiariate regularly arying on 0, 0, if, as t Z t, Z 2 ν in M + 0, 0,. t In such cases we hae Z, Z 2 CEV Mt, 0; t, 0; ν in 0, 0,. The aboe conergence implies that ν is homogeneous of order, that is, νcλ = c νλ for all c > 0 where Λ is a Borel subset of 0, 0,. By homogeneity arguments it follows that for r > 0, x ν{x, y 0, 0, : x + y > r, x + y Λ} = r ν{x, y 0, 0, : x + y >, =: r SΛ, x x + y Λ} where S is a measure on {x, y : x + y =, 0 x < }. The measure S is called the spectral measure corresponding to ν, while the measure ν is called the standardized measure. It was shown in Heffernan and Resnick 2007 that wheneer, Y CEV Mα, β; a, b; µ in, E γ with ψ x, ψ 2 x, 0, we hae the standardization f, f 2 Y CEV Mt, 0; t, 0; ν on the cone 0, 0,, for some

4 4 R. S. HAZRA AND K. MAULIK monotone transformations f and f 2. Das and Resnick 20 showed that this standardized measure ν is not a product measure. Throughout this article we assume that ψ x, ψ 2 x, 0 and consider the product of and Y. We remark that although the model can be standardized in this case, the standardization does not help one to conclude about the behaior of Y. 3. A brief oeriew of the results In this section we gie a brief description of the results in the subsequent sections. First note that if, Y CEV Mα, β; a, b; µ on, E γ, then α RV ρ and a RV γ, where α and a were the scalings for and Y respectiely. It need not a priori follow that DG ρ. We classify the problem according to the parameters γ and ρ. We break the problem into four cases depending on whether the parameters γ and ρ are positie or negatie. In Section 4 we show that depending on the properties of the scaling and centering parameters we can first reduce the basic conergence in conditional model to an equialent conergence with the limiting measure satisfying degeneracy condition in an appropriate cone. The reduction of the basic conergence helps us to compute the conergence of the product with ease in Section 5. Case I: ρ and γ positie: This is an easier case and the behaior is quiet similar to the classical multiariate extreme theory. In Theorem 5., we show that under appropriate tail condition on, the product Y has regularly arying tail of index /ρ + γ. It is not assumed that DG ρ, but in Section 6 we show that the tail condition is satisfied when DG ρ. It may happen that is in some other domain of attraction but still the tail condition holds. We also present a situation where the tail condition may fail. In all the remaining cases, at least one of the indices ρ and γ will be negatie. A negatie alue for γ will require that the upper endpoint of Y is indeed b. Howeer, as has been noted below, the same need not be true for ρ, and β. Yet, we shall assume that wheneer ρ < 0, the upper endpoint of the support of is β. Further, we shall assume at least one of the factors and Y to be nonnegatie. If both the factors take negatie alues, then, the product of the negatie numbers being positie, their left tails will contribute to the right tail of the product as well. In fact, it will become important in that case to compare the relatie heainess of the two contributions. This can be easily done by breaking each random ariable into its negatie and positie parts. For the product of two negatie parts, the releant model should be built on, Y and the analysis becomes same after that. While these details increase the bookkeeping, they do not proide any new insight into the problem. So we shall refrain from considering the situations where both and Y take negatie alues, except in Subcases IIb and IId below, where both and Y are nonpositie and we get some interesting result about the lower tail of the product Y easily. Case II: ρ and γ negatie: In Section 4, we first reduce the basic conergence to an equialent conergence where regular ariation can play an important role. In this case both bt and βt hae finite limit b and β respectiely. Since Y DG γ, b is the right end point of Y. Howeer, β need not be the right end point of in general, yet throughout we shall assume it to be so. In Section 4 we reduce the conditional model, Y CEV Mα, β; a, b; µ to, Ỹ which satisfies Conditions 2A and 2B with parameters α, 0; ã, 0; µ 2 on 0, 0,, where, = β and Ỹ = b Y, 3. and α and ã are some appropriate scalings and µ 2 is a transformed measure. Regular ariation at the right end point plays a crucial role during the determination the product behaior in this case,. Depending on the right end point, we break the problem into few subcases which are interesting. Subcase IIa: β and b positie: If the right end points are positie, then, without loss of generality, we assume them to be. In Theorem 5.2, we show that if and Y both hae positie right end point then Y has regularly arying tail of index / ρ, under some further sufficient moment conditions. In Section 7, we gie an example where the moment condition fails, yet the product shows the tail behaior predicted by Theorem 5.2. Subcase IIb: β and b zero: In Theorem 5.3 we show that if the both right end points are zero then the product conergence is a simple consequence of the result in Case I. In this case Y has regularly arying tail of index / ρ + γ.

5 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 5 Subcase IIc: β zero and b positie: We show in Theorem 5.4 that if Y is a nonnegatie random ariable haing positie right end point, then under some appropriate moment conditions Y has regularly arying tail of index / ρ. Subcase IId: β and b negatie: When both the right end points are negatie, then, without loss of generality, we assume them to be. In Theorem 5.5, we show that Y has regularly arying tail of index / ρ. There are a few more cases beyond the four subcases considered aboe, when both ρ and γ are negatie. For example, consider the case when Y has right end point zero and has positie right end point β. By our discussion aboe, should hae the support 0, β. Again, the product has right end point 0 and the behaior of around zero becomes important. Thus, to get something interesting in this case one must hae a conditional model which gies adequate information about the behaior of around the left end point. So it becomes natural to model, Y, which has already been considered in Subcase IIb. A similar situation occurs when β < 0 and b > 0. Here, again, the problem reduces to that in Subcase IId by modeling, Y. We refer to Remark 5. for a discussion on this subcase. Case III: ρ positie and γ negatie: In this case we assume b > 0 and also αt /at which implies that ρ = γ. We show in Theorem 5.6 that Y has regularly arying tail of index / γ. Case IV: ρ negatie and γ positie: In Theorem 5.7 we show that Y has regularly arying tail of index /γ. Finally we end this section by summarizing the results in a tabular form: Table. Behaior of products Index of α Index of a Theorem number Nature Regular ariation ρ > 0 γ > 0 Theorem 5. Y RV /γ+ρ ρ > 0 α a γ = ρ < 0 Theorem 5.6 Y RV / γ ρ < 0 γ < ρ Theorem 5.2 β b Y RV / ρ ρ < 0 α a γ = ρ Theorem 5.2 β b Y RV / ρ ρ < 0 γ < 0 Theorem 5.3 Y RV / γ + ρ ρ < 0 γ < 0 Theorem 5.4 Y RV / ρ ρ < 0 γ > 0 Theorem 5.7 Y RV /γ 4. Some transformations of CEVM according to parameters γ and ρ In this section we reduce the basic conergence in Condition 2A to an equialent conergence in some appropriate subspace of R 2 to facilitate our calculations of product of two ariables following conditional extreme alue model. We now discuss the four cases considered in Section 3. Case I: ρ and γ positie: In this case we assume Y is nonnegatie. Let, Y CEV Mα, β; a, b; µ on, E γ. Now by the domain of attraction condition 2. and Corollary.2.4 of de Haan and Ferreira 2006, we hae, bt at/γ, as t. Also, from Theorem 3..2 a,c of Bingham et al. 987 it follows that { βt lim t αt = 0 when ψ 2 = 0 ρ when ψ 2 0. Now using the aboe conditions and translating the and Y coordinates we get that, Y satisfies Conditions 2A and 2B with parameters α, 0; a, 0; ν on D =, 0,, for some nondegenerate measure ν which is obtained from µ by translations on both axes. So in Theorem 5. which deals with product Y in Case I, we assume that, Y satisfies Conditions 2A and 2B with parameters α, 0; a, 0; ν on D =, 0, for some nondegenerate Radon measure ν. Case II: ρ and γ negatie: Recall that in this case E γ =, γ. Since Y DG γ with γ < 0, it follows from Lemma.2.9 of de Haan and Ferreira 2006 that lim t bt =: b exists and is finite and

6 6 R. S. HAZRA AND K. MAULIK as t we hae, b bt at γ. Moreoer b turns out to be the right end point of Y. Hence in this case, without loss of generality we take at = γ b bt and it easily follows that, for y > 0, lim Ỹ t at > y = y γ, where Ỹ is defined in 3.. Now obsere that, Y CEV Mα, β; a, b; µ on, Eγ gies the, Ỹ which satisfy Conditions 2A and 2B with parameters α, β; at, 0; µ 2 on D, where, µ 2, x y, = µ, x γ y,. Now since ρ < 0, we get by Theorem B.22 of de Haan and Ferreira 2006 that lim t βt = β exists and is finite. It may happen that has a different right end point than β, but we assume β to be its right end point to aoid complications. In the coordinate we can do a similar transformation as the Y ariable, to get When ψ 2 0, we hae, as t, βt K t αtx + βt := P αt µ 2, x y, y γ β βt αt Now by Conergence of types Theorem, as t, we hae, Define, αt = { x b Y at =: Kx. K t ρ β βtx + βt Kx. ρ β βt when ψ 2 0 αt when ψ 2 = 0 > y ρ. 4. and ãt = at. 4.2 Using 4.2 we get, as t, { αt x, Ỹ ãt > y µ2, x + ρ y, for ψ 2 0 µ 2, x y, for ψ 2 = 0. In Section 5, we deal with Case II by breaking it up into different subcases as pointed out in Section 3. So, in Theorems , we assume that, Ỹ satisfy Conditions 2A and 2B with parameters α, 0; ã, 0; ν on 0, 0,, for some nondegenerate Radon measure ν. Case III: ρ positie and γ negatie: Since Y DG γ, we can do a transformation similar to that in Case II. So in this case, Ỹ satisfy Conditions 2A and 2B with parameters α, β; ã, 0; µ 3 on, 0,, for some nondegenerate measure µ 3. Now since ρ > 0, we can do a translation in the first coordinate to get, Ỹ which satisfy Conditions 2A and 2B with parameters α, 0; ã, 0; ν on, 0, for some nondegenerate measure ν. In Theorem 5.6 we deal with product in this case. Case IV: ρ negatie and γ positie: We assume that β is the right end point of and Y is nonnegatie. Now, as in Case II, we use 4. to get the following conergence for x 0 and y > 0, β Y βt x, αt at > y β βt Y = x +, αt αt at > y µ x +, y, as t. γ 4.3

7 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 7 So in Theorem 5.7, which deries the product behaior in Case IV, we assume β, Y satisfy Conditions 2A and 2B with parameters α, 0; a, 0; ν on 0, 0,, for some nondegenerate Radon measure ν. We thus obsere that if, Y CEV Mα, β; a, b; µ, then in Cases I, II, III and IV respectiely,, Y,, Ỹ,, Ỹ and β, Y satisfy Conditions 2A and 2B with some positie scaling parameters, zero centering parameters and a nondegenerate limiting Radon measure on D =, 0,. In future sections, wheneer we refer to Conditions 2A and 2B with respect to the transformed ariables alone without any reference to the CEVM model for the original pair, Y, we shall denote, by an abuse of notation, the limiting Radon measure for the transformed random ariables as µ as well. 5. Behaior of the product under conditional model Now we study the product behaior when, Y CEV Mα, β; a, b; µ. In all the cases we assume Conditions 2A and 2B on the suitably transformed ersions of, Y, so that centering is not required. Case I: ρ and γ positie: We begin with the case where both ρ and γ are positie. As mentioned earlier, we assume γ > 0. Theorem 5.. Let ρ > 0, γ > 0 and Y be a nonnegatie random ariable. Assume, Y satisfy Conditions 2A and 2B with parameters α, 0; a, 0; µ on D :=, 0,. Also assume lim lim sup ɛ 0 αt > z = ɛ t Then, Y has regularly arying tail of index /γ + ρ and as t, Y/αtat conerges aguely to some nondegenerate Radon measure on, \ {0}. Proof. For ɛ > 0 and z > 0 obsere that the set, A ɛ,z = {x, y D : xy > z, y > ɛ} is a relatiely compact set in D and µ is a Radon measure. Hence there exists a sequence ɛ k 0 such that µ A ɛk,z = 0 for all k. For the lower bound, we hae, Y Y lim inf t αtat > z lim inf t αtat > z, Y at > ɛ k = µa ɛk,z. Hence as k, µa ɛk,z µ{x, y D : xy > z}. Also, Y Y lim sup t αtat > z lim sup t αtat > z, Y at > ɛ k Y + lim sup t αtat > z, Y at ɛ k µa ɛk,z + lim sup t αt > z ɛ k Now due to the gien tail condition 5. on, as k we get, Y lim sup t αtat > z µ{x, y D : xy > z}. By similar arguments one can show the aboe conergence on the sets of the form, z also. Spectral form for product: For simplicity let us assume that is nonnegatie as well. Then the ague conergence in M + D can be thought of as ague conergence in M + 0, 0,. By Theorem 5. we hae, Y lim t αtat > z = µ{x, y 0, 0, : xy > z}. Since ρ > 0 and γ > 0 it is known that there exists αt αt and at at such that they are eentually differentiable and strictly increasing. Also αtx x ρ and atx x γ as t. Hence, α t x, a Y t αt > y = at αt αtx αt, Y at > aty at

8 8 R. S. HAZRA AND K. MAULIK µ0, x ρ y γ, = µt 0 0, x y,, where T 0 x, y = x /ρ, y /γ. Let S be the spectral measure for the standardized pair α, a Y corresponding to µt 0. Then, µ{x, y 0, 0, : xy > z} =µt0 {x, y 0, 0, : x ρ y γ > z} = r 2 drsdω = ω 0, ω 0, =z ρ+γ r ρ+γ ω ρ ω γ >z z ρ+γ r> ω ρ ω γ ρ+γ ω 0, r 2 drsdω ω ρ ρ+γ ω γ ρ+γ Sdω. So finally we hae, Y lim t αtat > z = µ{x, y 0, 0, : xy > z} = z ρ+γ ω ρ γ ρ+γ ω ρ+γ Sdω. ω 0, Case II: ρ and γ negatie: As has been already pointed out, β need not be the right endpoint. Howeer, we shall assume it to be so. The tail behaior of Y strongly depends on the right end points of and Y. There are seeral possibilities that may arise, but it may not always be possible to predict the tail behaior of Y in all the cases. We shall deal with few interesting cases. Regarding one of the cases left out, see Remark 5.. Recall that, from 3. we hae, β b Y = Ỹ β + b Ỹ 5.2 Subcase IIa: β and b positie: After scaling and Y suitably, without loss of generality, we can assume that β = = b. Theorem 5.2. Suppose and Y are nonnegatie and, Ỹ satisfies Conditions 2A and 2B with parameters α, 0; ã, 0; µ on 0, 0,. Assume E / ρ +δ < for some δ > 0 and either γ < ρ or αt/ãt remains bounded. Then Y has regularly arying tail of index / ρ and, as t, Y / αt conerges aguely to some nondegenerate Radon measure on 0,. We start with a technical lemma. Lemma 5.. For 0 t t 2 and z > 0 we denote the set V t,t 2,z = {x, y 0, 0, : x t, t 2, y > z}. 5.3 Suppose that {Z t, Z 2t } is a sequence of pairs of nonnegatie random ariables and there exists a Radon measure ν in 0, 0, such that they satisfy the following two conditions. Condition A: Wheneer V t,t 2,z is a ν-continuity set, we hae, as t, Condition B: For any z 0 0, we hae as t, Then, as t, Z t, Z 2t V t,t 2,z νvt,t 2,z. Z t, Z 2t V 0,,z0 fz0 0,. Z t, Z 2t ν

9 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 9 in M + 0, 0, Proof. Fix a z 0 0, and define the following probability measures on 0, z 0, : Q t = Z t, Z 2t and Q = ν Z t, Z 2t V 0,,z0 fz 0. From condition A it follows that Q t V t,t 2,z QV t,t 2,z wheneer ν V t,t 2,z = 0 as t. Now following the arguments in the proof of Theorem 2. of Maulik et al. 2002, it follows that Q t weakly conerge to Q on 0, z 0,. Since a Borel set with boundary haing zero Q measure is equialent to haing measure zero with respect to the measure ν we hae that Z t, Z 2t B νb for any Borel set haing boundary with zero ν measure. Let K be a relatiely compact set in 0, 0, with ν K = 0. Then there exists z 0 > 0 such that K 0, z 0,. Then B is Borel in 0, z 0, and ν B = 0. Hence we hae, Z t, Z 2t K = Z t, Z 2t K νb = νk. This shows that Z t, Z 2t aguely conerges to ν on 0, 0,. From 5.2, the behaior of Y will be determined by the pair Ỹ, + Ỹ. So we next proe a result about the joint conergence of product and the sum. Lemma 5.2. Let γ < 0, ρ < 0 and, Ỹ satisfies Conditions 2A and 2B with parameters α, 0; ã, 0; µ. If E / γ +δ < for some δ > 0, then Ỹ, + Ỹ also satisfies Conditions 2A and 2B with parameters αã, 0; ã, 0; µt 2 on 0, 0, where T 2 x, y = xy, y. Proof. First obsere that from the compactification arguments used in Lemma 5. and basic conergence in Condition 2A satisfied by the pair, Ỹ, it follows that Ỹ αtãt, Ỹ µt2 in M + 0, 0,. 5.4 ãt Let 0 t t 2 and z > 0. Assume that µt2 V t,t 2,z = 0, where V t,t 2,z is defined in 5.3. Since is nonnegatie, is greater than or equal to and hence for a lower bound we get, lim inf t Ỹ αtãt, + Ỹ ãt V t,t 2,z lim inf t Ỹ αtãt t Ỹ, t 2, ãt > z = µt2 V t,t 2,z. 5.5 For the upper bound, choose 0 < ɛ < z, such that / ãt < ɛ/2 since ãt RV γ and V t,t 2,z ɛ is a continuity set for the measure µt2. Then, Ỹ αtãt, + Ỹ V t,t ãt 2,z Ỹ αtãt, + Ỹ V t,t ãt 2,z, ãt ɛ k 2 Ỹ + αtãt, + Ỹ V t,t ãt 2,z, ãt > ɛ k 2 Ỹ αtãt t Ỹ, t 2, ãt > z ɛ k + ãt > ɛ k 2 Ỹ αtãt, Ỹ V t,t2,z ɛk ãt E / γ +δ + 2 / γ +δ t ãtɛ k γ +δ.

10 0 R. S. HAZRA AND K. MAULIK The first term conerges to µt2 V t,t 2,z ɛ k, while the second sum conerges to zero, since ãt RV γ. Now letting ɛ 0 satisfying the defining conditions, we obtain the upper bound, which is same as the lower bound 5.5. Thus we get that, lim t Ỹ αtãt, + Ỹ ãt V t,t 2,z = µt 2 V t,t 2,z. Hence Condition A of Lemma 5. is satisfied. Now if we fix z 0 0, and let ɛ > 0 satisfy the conditions as in the upper bound aboe, then Ỹ αtãt, + Ỹ + Ỹ V 0,,z0 = > z 0 ãt ãt Ỹ ãt > z 0 ɛ + ãt > ɛ 2 z 0 ɛ γ. Hence the upper bound for the required limit in Condition B of Lemma 5. follows by letting ɛ 0. The lower bound easily follows from the domain of attraction condition on Y and the fact that. So Ỹ +Ỹ Condition B is also satisfied by the pair αtãt, ãt and hence the result follows from Lemma 5.. Proof of Theorem 5.2. Denote W = Ỹ, W = + Ỹ and note that Y = W /W. So from preious lemma it follows that, W αtãt, W µt2 as t. ãt Let w 0, and ɛ > 0 and consider the set, Then, for ɛ > 0, we hae, W W αt > w A w,ɛ = {x, y 0, 0, : x > yw, y > ɛ}. 5.6 W = αtãt, W W W A w,ɛ + ãt W > w, αt ãt ɛ. Since the set is bounded away from both the axes, the first sum conerges to µt2 A w,ɛ by the ague conergence of W, W. Now since 0 we hae + Ỹ Ỹ, and hence for large t we get, Y > w, Ỹ ɛ k αt ãt Y > w αt, Y ãtɛ k + ãtɛ > k + Ct, w, k E w αt ãtɛ k + ρ +δ, where, Ct, w, k = t αt ρ +δ ρ +δ ãtɛ k + w ãt αt ɛ k + which goes to zero as t, since αt RV ρ and αt/ãt remains bounded. αt ρ +δ, Subcase IIb: β = 0 and b = 0: In this case, both and Y are nonpositie, but the product Y is nonnegatie. Thus, the right tail behaior of Y will be controlled by the left tail behaiors of and Y, which we cannot control much using CEVM. Howeer, CEVM gies some information about the left tail behaior of Y at 0, which we summarize below.

11 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL Note that in this case, from 3. and 4.2, we hae = /, Ỹ = /Y and αt = / ρ βt, ãt = /at. From Theorem 5., the behaior of the product Y around zero, or equialently the behaior of the reciprocal of the product Ỹ around infinity, follows immediately. Theorem 5.3. If ρ < 0, γ < 0 and, Ỹ satisfy Conditions 2A and 2B with parameters α, 0; ã, 0; µ. Also suppose, lim lim sup ɛ 0 t αt > z = 0 ɛ Then Y has regularly arying tail with index / γ + ρ and as t, Y / αtãt conerge to some nondegenerate Radon measure on 0,. Subcase IIc: β = 0 and b = : Now note that from 5.2 we hae, Y = Ỹ Ỹ. Theorem 5.4. Suppose Y is nonnegatie and, Ỹ satisfy Conditions 2A and 2B with parameters α, 0; ã, 0; µ. If E / ρ +δ < for some δ > 0, then Y has regularly arying tail of index / ρ and as t, Y / αt conerge aguely to some nondegenerate measure on 0,. Proof. First obsere that under the hypothesis of the Theorem the conergence 5.4 holds and as ãt, it can be shown by arguments similar to Lemma 5.2 that, as t Ỹ αtãt, Ỹ µ, 5.7 αt in M + 0, 0,, for some nondegenerate Radon measure µ. Now for z > 0 and ɛ > 0 consider the set A z,ɛ as in 5.6 and note, Ỹ αtãt, Ỹ Y A z,ɛ αt αt > z Ỹ αtãt, Ỹ αt A z,ɛ + Ỹ αtỹ Ỹ > z, ãt The term on the left hand side and the first term on the right hand side conerge due the ague conergence 5.7. For the second term on the right hand side obsere that, Ỹ Ỹ Y > z, ɛ > z, αtỹ ãt αt Ỹ ãtɛ + Y > z, Y αt ãtɛ + Y αt > z, Y ãtɛ + < αt/z ãtɛ+ αt/z t E / ρ +δ ãtɛ+ ρ +δ ɛ. The last expression tends to zero as ãt and α RV ρ.

12 2 R. S. HAZRA AND K. MAULIK Subcase IId: β and b negatie: As in Subcase IIa, after suitable scaling, without loss of generality, we can assume that β = b =. Again, from 5.2, we hae Y = Ỹ + Ỹ + and to get the behaior of the product around we first need to derie the joint conergence of Ỹ, + Ỹ +. Using an argument ery similar to Theorem 5.2, we immediately obtain the following result. Theorem 5.5. Let, Ỹ satisfy Conditions 2A and 2B with parameters α, 0; ã, 0; µ and E ρ +δ < for some δ > 0. If either γ < ρ or αt/ãt remains bounded, then Y has regularly arying tail of index / ρ and as t, Y / αt conerges aguely to some nondegenerate Radon measure on 0,. Remark 5.. The other case when β = and b = 0 is not easy to derie from the informations about the conditional extreme alue model. In this case the right endpoint of the product is zero and behaior of around zero seems to be important. But the conditional model gies us the regular ariation behaior around one and not around zero. Case III: ρ positie and γ negatie: In this case we shall assume that is nonnegatie and the upper endpoint of Y, b is positie. If b 0, then the behaior of around its lower endpoint will play a crucial role in the behaior of the product Y, which becomes negatie. Howeer, the behaior of around its lower endpoint is not controlled by the conditional model and we are unable to conclude about the product behaior when b 0. So we only consider the case b > 0. We also make the assumption that αt ãt, which requires that ρ = γ. Theorem 5.6. Let be nonnegatie and Y has upper endpoint b > 0. Assume that, Ỹ satisfy Conditions 2A and 2B with parametersα, 0; ã, 0; µ with αt at and E / γ +δ < for some δ > 0, then Y has regularly arying tail of index / γ and as t, we hae Y /ãt conerges aguely to a nondegenerate Radon measure on 0,. Proof. As αt ãt conergence of types allows to change αt to at = ãt and hence, Ỹ satisfy Conditions 2A and 2B with parameters ã, 0; ã, 0; µ. Using the fact Ỹ = b Y, we hae Y = b Ỹ. Ỹ Using arguments similar to Proposition 4 of Heffernan and Resnick 2007, it can be shown that as t, Ỹ, b Ỹ µt3, ãt in M + 0, 0,, where T 3 x, y = x/y, b y. Since ãt, we further hae, as t, Ỹ, b Ỹ µt3. ãt Now applying the map T 2 x, y = xy, y to the aboe ague conergence and using compactification arguments similar to that in the proof of Theorem 2. of Maulik et al. 2002, we hae, Ỹ b Ỹ, b Ỹ ãt ãt µt 3 T 2. Recalling the facts that Y = b Ỹ /Ỹ and ãt and reersing the arguments in the second coordinate, we hae, Y ãt, Ỹ µt3 T2 T 3, 5.8 ãt

13 where T 3 x, y = x, hae following series of inequalities, Y ãt > z, since ã RV γ. Now obsere that Y ãt > z, Ỹ ãt > ɛ PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 3 y b. Obsere that µ is a Radon measure on 0, 0,. If ɛ > 0 and z > 0 we Ỹ ãt ɛ Y = Y = ãt ãt > z, b Y ãtɛ > z, Y b ãtɛ ãt > z b ãtɛ E +δ γ t b γ +δ 0, ãtz γ +δ ãtɛ Y ãt > z Y ãt > z, Ỹ ãt > ɛ Y + ãt > z, Ỹ ãt ɛ. The last term on the right side is negligible by the preious argument and the the term on the left side and the first term on the right side conerges due to the ague conergence in 5.8. The result then follows by letting ɛ 0. Remark 5.2. Theorem 5.6 requires that / γ + δ-th moment of is finite. Howeer, this condition is not necessary. In the final section we gie an example where this moment condition is not satisfied but we still obtain the tail behaior of the product. Case IV: ρ negatie and γ positie: In this case we shall assume that Y is nonnegatie, β > 0 and β is the upper endpoint of. Arguing as in Case III, we neglect the possibility that β 0. Thus we further assume 0 β. Since becomes bounded, the product of Y inherits its behaior from the tail behaior of Y. Theorem 5.7. Assume that both and Y are nonnegatie random ariables with β being the upper endpoint of. Let β, Y satisfy Conditions 2A and 2B with parameters α, 0; a, 0; µ on 0, 0,, for some nondegenerate Radom measure µ Then Y has regularly arying tail of index /γ and for all z > 0, we hae Y at > z z γ β γ, as t. Proof. First we proe the upper bound which, in fact, does not use the conditional model. Fix ɛ > 0 and obsere that Y Y Y at > z = > z, ɛ < < β + at at > z, ɛ Y at > z Y + β at > z ɛ z γ z γ + as t. β ɛ The second term goes to zero as one allows ɛ to go to zero. To proe the lower bound we use the basic conergence in Condition 2A for the pair β, Y. Before we show the lower bound first obsere that by arguments similar to the proof of Theorem 2. of Maulik et al we hae, β Y, αtat Y at µt 2 as t 5.9

14 4 R. S. HAZRA AND K. MAULIK in M + 0, 0,, where recall that T 2 x, y = xy, y. Now to show the lower bound, first fix a large M > 0 and ɛ > 0 such that αt < ɛ for large t recall that α RV ρ and αt 0 in this case. Now note that Y at > z β Y = αtat β Y αt + β Y at > z M, β Y > z + Mαt αtat at β Y M, β Y > z + Mɛ αtat at z + Mɛ µt2 0, M β,, using 5.9. First letting ɛ 0 and then letting M, so that 0, M z/β, are continuity sets of the measure µt 2, we get that lim inf Y > z µt t 2 0, z/β, = z γ β γ. at 6. Some remarks on the tail condition 5. We hae already noted that the compactification arguments in Section 5 require some conditions on the tails of associated random ariables. Except in Theorem 5., they hae been replaced by some moment conditions, using Marko inequality. The tail condition 5. in Theorem 5. can also be replaced by the following moment condition: If for some δ > 0, we hae E /ρ+δ <, then 5. holds, as α RV ρ. In general, if, Y follows CEVM model, it need not be true that DG ρ. Howeer, DG ρ with scaling and centering functions αt and βt and 0, then the moment condition 5.. In fact, lim sup t αt > z ɛ is a constant multiple of z/ɛ /ρ, which goes to zero as ɛ 0. The tail condition 5. continues to hold in certain other cases as well. Suppose that 0 and DG λ with scaling and centering At and Bt and λ < ρ. In this case, α RV ρ and A RV λ. Thus, αt/at. Hence, for any ɛ > 0, we hae αt > z = ɛ At > zαt ɛat is of order of αt/at /λ, which goes to zero and 5. holds. Howeer, it would be interesting to see the effect of A and B as scaling and centering in CEVM model. Since, α is of an order higher than A, the limit, as expected, becomes degenerate. Proposition 6.. Let the pair, Y CEV Mα, β; a, b; µ with ρ > 0 and γ > 0. Assume DG λ with ρ > λ and centering and scaling as At and Bt. If Bt lim x, Y bt > y t At at exists for all continuity points x, y R E γ, then, for any fixed y E γ, as x aries in R, the limit measure assigns same alues to the sets of the form, x y,. Proof. Obsere that Therefore we hae, Bt lim t At At Bt βt x + = At αt αt αt x, Y bt at > y x + Bt βt At αt 0 x + λ ρ.

15 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 5 βt = lim t αt At αt =µ, /ρ y,. Bt βt x +, Y bt > y αt at We would also like to consider the remaining case, namely, when 0 and DG λ with scaling A and λ > ρ. Clearly, we hae αt/at 0. Thus, for any ɛ > 0, we hae αt > z = ɛ At > zαt ɛat Hence 5. cannot hold and Theorem 5. is of no use. The next result show that in this case we hae multiariate extreme alue model with the limiting measure being concentrated on the axes, which gies asymptotic independence. Theorem 6.. Let, Y CEV Mα, 0; a, 0; µ on the cone 0, 0,, DG λ with λ > ρ and scaling At RV λ. Also assume that DG λ with λ > ρ. At αt. Then At, Y 6. at conerges aguely to a nondegenerate Radon measure on 0, 2 \ {0, 0}, which is concentrated on the axes. Proof. We first show the ague conergence on 0, 2 \ {0, 0}. Let x > 0, y > 0. Then, At, Y Y 0, x 0, y c = at At > x + at > y At > x, Y at > y. The first two terms conerge due to the domain of attraction conditions on and Y. For the last term, by the CEVM conditions and the fact that At/αt, we hae At > x, Y at > y = αt > xat αt, Y at > y This establishes 6.. Howeer, using x = y in 6.2, we find that the limit measure does not put any mass on 0, 2. Remark 6.. Since, we hae asymptotic independence, seeral different behaiors for the product are possible, as it has been illustrated in Maulik et al While we hae established asymptotic independence on the larger cone 0, 2 \ {0} in Theorem 6., CEVM gies another nondegenerate limit µ on the smaller cone 0, 0,. Thus,, Y exhibits hidden regular ariation, as described in Maulik and Resnick 2004, Resnick Example We now consider the moment condition in Theorem 5.2. We show that the condition is not necessary by proiding an example, where the condition fails, but we explicitly calculate the tail behaior of Y. Let and Z be two independent random ariables, where follows Beta distribution with parameters and a and Z be supported on 0, and be in DG /b, for some a > 0 and b > 0. Thus, we hae P > x = x a and PZ > x = x b Lx for some slowly arying function L. Let G denote the distribution function of the random ariable Y = Z. Then Gx = x a+b L x and hence Y DG /a+b. Clearly, for = /, we hae E a+b = and the moment condition in Theorem 5.2 fails for ρ = /a + b. For a nondecreasing function H, we define the left continuous inerse see, for example, Resnick, 987, Section 0.2 as H x = inf{s : Hs x}. Using this definition, we further define ãt = /G t.

16 6 R. S. HAZRA AND K. MAULIK Since, Y DG /a+b, we hae from Corollary.2.4 of de Haan and Ferreira 2006, Then, for x > y > 0, we hae, ãt x, ãt a+b t. 7. Lãt Ỹ ãt > y = ãt x, ãty > y = yãt < xãt, Z > yãt = t ãt a+b y a x a y b Lyãt tlãt ãt a+b y a x a y b y a x a y b, where we use 7. in the last step. Thus,, Ỹ satisfies Conditions 2A and 2B with ρ = γ = /a+b. Finally, we directly calculate the asymptotic tail behaior of the product. For simplicity of the notations, for y > 0, let us denote ct = /ãty. Since ã RV /a+b, as t, we hae ct. Also, ãt = / cty. Then, Y > ãty =a t P Z > ct s a ds ct s a ã cty a ã cty substituting s = z ct, ct P =a ct a+b ã cty a ct a+b ã cty Z > c s a ds s c b s ct s a L + ct 0 L ct ct ct/s ds z b ctz b za L /2 z b z a dz, 0 + ct dz ct z where, in the last step, we use Dominated Conergence Theorem and the facts that ct and ct ct ct L + L L ct z ct z ct uniformly on bounded interals of z. Finally, using 7., definition of ct, to get, Y /2 > y a y a+b z b z a dz. ãt 0 References N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular ariation, olume 27 of Encyclopedia of Mathematics and its Applications. Cambridge Uniersity Press, Cambridge, 987. ISBN B. Das and S. Resnick. Conditioning on an extreme component: Model consistency and regular ariation on cones. Bernoulli, 7: , 20. Laurens de Haan and Ana Ferreira. Extreme alue theory. Springer Series in Operations Research and Financial Engineering. Springer, New York, ISBN ; An introduction.

17 PRODUCTS IN CONDITIONAL ETREME VALUE MODEL 7 Janet E. Heffernan and Sidney I. Resnick. Limit laws for random ectors with an extreme component. Ann. Appl. Probab., 72:537 57, ISSN Janet E. Heffernan and Jonathan A. Tawn. A conditional approach for multiariate extreme alues. J. R. Stat. Soc. Ser. B Stat. Methodol., 663: , ISSN With discussions and reply by the authors. Anthony W. Ledford and Jonathan A. Tawn. Statistics for near independence in multiariate extreme alues. Biometrika, 83:69 87, 996. ISSN Anthony W. Ledford and Jonathan A. Tawn. Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B, 592: , 997. ISSN Krishanu Maulik and Sidney Resnick. Characterizations and examples of hidden regular ariation. Extremes, 7: , ISSN Krishanu Maulik, Sidney Resnick, and Holger Rootzén. Asymptotic independence and a network traffic model. J. Appl. Probab., 394:67 699, ISSN Sidney Resnick. Hidden regular ariation, second order regular ariation and asymptotic independence. Extremes, 54: , ISSN S. I. Resnick. Extreme alues, regular ariation, and point processes, olume 4 of Applied Probability. A Series of the Applied Probability Trust. Springer-Verlag, New York, 987. ISBN Rajat Subhra Hazra, Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 70008, India address: rajat r@isical.ac.in Krishanu Maulik, Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 70008, India address: krishanu@isical.ac.in