CHARACTERIZATIONS AND EXAMPLES OF HIDDEN REGULAR VARIATION

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1 CHARACTERIZATIONS AND EXAMPLES OF HIDDEN RULAR VARIATION KRISHANU MAULIK AND SIDNEY RESNICK Abstract. Random vectors on the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn 996, 997. We provide characterizations and examples of such distribution in terms of mixture models and product models.. Introduction. A refinement of the class of multivariate regularly varying distributions, which we call hidden regular variation, is a semi-parametric subfamily of the full family of distributions possessing multivariate regular variation and asymptotic independence. Various cases of hidden regular variation have received considerable attention recently as part of the program to statistically distinguish asymptotic independence from dependence. See Campos et al. 22, Coles et al. 999, de Haan and de Ronde 998, Draisma et al. 2, Ledford and Tawn 996, 997, Peng 999, Poon et al. 23, Resnick 22a, Stărică 999, 2. In particular, hidden regular variation is based on the analysis of the coefficient of tail dependence of Ledford and Tawn 996, 997. Both multivariate regular variation and hidden regular variation are crucial to defining semi-parametrically specified dependence structures for heavy tailed models where it is important to consider assumptions beyond independence. These concepts enter into analyses because there are often patterns of dependence among large values of the components of a random vector which are not apparent by using summaries of dependence based on the center of the distributions such as cross correlations. See de Haan and de Ronde 998, Resnick 22b,c. A treatment of hidden regular variation, placing it in relation to the concepts of asymptotic independence and second order regular variation, was given in Resnick 22b where examples and characterizations were given. Here we continue discussion by providing further characterizations and canonical representations. The characterizations when the hidden angular measure is infinite are not completely satisfactory. These canonical representations are dependent on a notion of multivariate tail equivalence. We also discuss how influential hidden regular variation can be in assessing the behavior of products of components of a random vector possessing hidden regular variation... Outline. The rest of this section reviews notation Subsection.2 and the polar coordinate transformation Subsection.3. Section 2 defines multivariate regular variation of a probability distribution on R d +, d > and we phrase our definitions in terms of vague convergence of measures. The orientation to measures is most natural in a multivariate context where d > and when considering changes of coordinate systems. Section 2 also defines precisely hidden regular variation. Section 3 defines a notion of tail equivalence for multivariate distributions. The one dimensional concept, considered in Resnick 97, has a natural definition which says that if two distributions F, G are tail equivalent, meaning for some c,, F x cgx, as x, then extremal behavior of i.i.d. samples from F is essentially the same as for i.i.d. samples from G. Here and afterwards, F x := F x denotes the complementary probability distribution function corresponding to F. How to define a multivariate analogue Key words and phrases. heavy tails, regular variation, Pareto tails, coefficient of tail dependence, hidden regular variation, asymptotic independence. Sidney Resnick s research was partially supported by NSF grant DMS and NSA grant MSPF-2G-83 at Cornell University.

2 2 K. MAULIK AND S. RESNICK depends on the purpose to which the concept will be put. We propose a definition and based on this definition, we characterize distributions with hidden regular variation with finite hidden angular measure as tail equivalent to certain mixtures. In Section 4, we give a characterization and representation of distributions with finite hidden angular measure. Section 5, on the other hand, considers the case of infinite hidden angular measure. We give a characterization for this case and several important examples but the characterization has deficiencies. Finally, in Section 6, we study the influence of hidden regular variation on the behavior of the distribution of the product of two asymptotically independent random variables..2. Notation. For simplicity we will generally assume that random vectors have non-negative components. Set E :=, d \ {} so that the origin =,..., is excluded from E. Compact subsets of E are compact sets of, d which do not intersect the origin; see the discussion in Resnick 22b. In some applications, for instance in finance, it is natural to consider the cone, d \ {}. We leave it to the reader to make the modest changes necessary to generalize to this case, by considering the orthants individually. Vectors are denoted by bold letters, capitals for random vectors and lower case for non-random vectors. For example: x = x,..., x d R d. Operations between vectors should be interpreted componentwise so that for two vectors x and z, x < z means x i < z i, i =,..., d, zx = z x,..., z d x d, and so on. For a real number c, denote as usual cx = cx,..., cx d. We denote the rectangles or the higher dimensional intervals by a, b = {x E : a x b}. Higher dimensional rectangles with one or both endpoints open are defined analogously. Complements are taken with respect to E, so that for x >, { } d, x c y i = E \, x = y E : x >. i For i =,..., d, define the basis vectors i= e i =,...,,,,..., so that the axes originating at are L i := {te i, t > }, i =,..., d. Then define the cone E = E \ d i= L i. The cone E consists of points of E such that at most d 2 coordinates are. We further define the subcone E =,, the cone consisting of points of E with only non-zero coordinates. If d = 2, we have E = E..3. The polar coordinate transformation. It is sometimes illuminating to consider multivariate regular variation for the distribution of a random vector after a polar coordinate transformation. Suppose that : R d, is a norm on R d. The most useful norms for us are the usual Euclidean L 2 norm, the L p norm for p > and the L norm: x = n i= xi. Assume the norm has been scaled so that e i = for i =,..., d. Given a chosen norm, the points at unit distance from the origin are ℵ := {x E : x = }. For the purpose of hidden regular variation, we need to look at a smaller subcone C of E and the restriction of ℵ to C is denoted by ℵ C = ℵ C. Recall norms on R d are all topologically equivalent in that convergence in one norm implies convergence in another. Now fix a norm. Define the polar coordinate transformation T :, \ {}, ℵ by T x = x, x x =: r, a, and the inverse transformation T :, ℵ, \ {} by T r, a = ra.

3 HIDDEN RULAR VARIATION 3 Think of a ℵ as defining a direction and r telling how far in direction a to proceed. Since we excluded from the domain of T, both T and T are continuous bijections. When d = 2, it is customary, but not obligatory, to write T x = r, θ, where x = r cos θ, r sin θ, with θ π/2, rather than the more consistent notation T x = r, cos θ, sin θ. For a random vector X in R d we sometimes write T X = R X, Θ X. When there is no risk of confusion, we suppress the subscript. 2. Multivariate regular variation and hidden regular variation. 2.. Multivariate regular variation for distributions of random vectors. Multivariate regular variation is usually either defined by convergence of a family of functions defined on the cone C E or as vague convergence of measures defined on the Borel subsets of the cone. Behavior on the boundary of the cone can be crucial. See Balkema 973, Basrak et al. 22, Davis and Hsing 995, Davis and Mikosch 998, de Haan et al. 984, Meerschaert and Scheffler 2, Resnick 986, Resnick 987, Chapter 5. The cones most commonly used are E itself and E. Consider a d-dimensional non-negative random vector Y = Y,..., Y d whose range includes the cone C. We suppose that all one-dimensional marginal distributions are the same. This can be achieved by marginal transformations of the components. While this method of transformation is a simple theoretical procedure, associated statistical difficulties should not be neglected. The distribution of Y is multivariate regularly varying on the cone C with limit measure ν, a non-zero Radon measure defined on Borel subsets of C, if there exists a function bt as t, such that 2. Y bt in M + C, the space of positive Radon measures on C. Here convergence is vague convergence of measures and M + C has the vague topology. See, for example, Kallenberg 983, Karr 986, Neveu 977, Resnick 987. Condition 2. implies that there exists a non-negative constant α such that for relatively compact sets B C v ν 2.2 νcb = c α νb, c >, which is called the homogeneity property of ν. Henceforth, assume that α >. The function b is necessarily regularly varying of index /α which we write as b RV /α and we can choose b to be strictly increasing, and continuous. We shall always assume that b is strictly continuous and increasing. There are equivalent formulations in terms of polar coordinates cf. Basrak, 2, Basrak et al., 22, Resnick, 22b,c. The regular variation condition 2. is equivalent to 2.3 R bt, Θ where c >, ν α is a Radon measure on, given by v c ν α S, ν α x, = x α, x >, S is a Radon measure on ℵ C and can be chosen to be a probability measure if ℵ C is compact, and convergence is in M +, ℵC. Asymptotic independence occurs when C = E and νe = or equivalently when S concentrates uniformly on the set {e i, i d} Hidden regular variation for distributions of random vectors. The random vector Y has a distribution possessing hidden regular variation if there is a subcone C C and the distribution has regular variation on C but also regular variation on C with scaling function of lower order. So in addition to 2., we assume that there exists a non-decreasing function b t such that bt/b t and there exists a measure ν which is Radon on C and such that Y 2.4 b t v ν

4 4 K. MAULIK AND S. RESNICK on M + C. Then there exists α α such that b RV /α and ν and α satisfy the analogue of 2.2. We can again assume that b is strictly increasing, continuous and further assume that b = and hence b =. When C = E and C = E, then cf. Resnick, 22b hidden regular variation implies asymptotic independence. In this case, the name hidden regular variation is justified by the fact that the relatively crude normalization necessary for convergence on the axes is too large and obscures structure in the interior of the cone. A normalization of smaller order, b, is necessary on the smaller cone. Also, as has previously been noted by several authors including de Haan and de Ronde 998, pg. 4, Resnick 987, pg. 297, Sibuya 96, asymptotic independence can be quite different from independence. The measure ν of hidden regular variation is a measure of the dependence in asymptotic independence. There are possible variants of hidden regular variation. First, the normalization in 2.4 of each component of the random vector is by the same function. One could allow different normalizations b i t, i =,..., d, provided bt/b i t for i =,..., d. The case of different normalizing functions can be reduced to 2.4 by monotone transformations. Second, the definition of hidden regular variation uses two cones C C. In principle, one could have more cones C C C C k with regular variation of progressively smaller order present in each. A simple example where d = 3 = k is given in Example 5. below. A characterization of hidden regular variation when C, C = E, E is given in Resnick 22b. It uses max- and min-linear combinations of the form d i= s iy i and d i= a iy i. See also the work of Coles et al The hidden regular variation condition 2.4 has an equivalent form in polar coordinates: R 2.5 b t, Θ v c ν α S, where c >, S is a Radon measure on ℵ := ℵ C and convergence is in M +, ℵ. The measure ν in 2.4 can be either finite or infinite on {x C : x > } and equivalently, the measure S in 2.5 can be either finite or infinite on ℵ. See Examples and 2 in Resnick 22b. We will call the measure S the angular measure or spectral measure. S is called the hidden angular measure or hidden spectral measure. Keep in mind these are quantities defined by an asymptotic procedure. The coefficient of tail dependence is defined by Ledford and Tawn 996, 997, 998 and is a somewhat weaker concept than our definition of hidden regular variation. Schlather 2 points out some curiosities stemming from the formulation. When hidden regular variation is present, the coefficient of tail dependence is α. Related work is in Coles et al. 999, reviewed and discussed more fully in Heffernan 2, where two limits χ, χ are assumed to exist in the context d = 2. When hidden regular variation is assumed as in 2. and 2.4, we have χ := lim t P Y 2 > t Y > t = 2 log P Y > t χ := lim t log P Y > t, Y 2 > t = 2α α. α 3. Multivariate tail equivalence. Two probability distributions F, G on R + are tail equivalent see Resnick, 97 if they have the same right endpoint x and for some c,, Gs lim s x F s = c. Then, upper extremes of i.i.d. samples from F behave asymptotically in the same way as upper extremes of i.i.d. samples from G. In the heavy tailed case, this definition is equivalent to supposing that there exists a function bt such that for some α >, on,, as t tf bt v ν α, tgbt v cν α.

5 HIDDEN RULAR VARIATION 5 Now suppose Y and Z are, -valued random vectors with distributions F, G respectively. In the multivariate regular variation context, we say that F and G or by abuse of language Y and Z are tail equivalent on the cone C E if there exists a scaling function bt such that Y 3. bt = tf bt v Z ν and bt = tgbt v cν in M + C, for some constant c, and non-zero Radon measure ν on C. We shall write Y tec Z. If {Y n, n } is an i.i.d. sample from F and {Z n, n } is an i.i.d. sample from G, then n n 3.2 ɛ Y i/bn PRM ν, and ɛ Zi/bn PRM cν, i= in M p C, the space of Radon point measures on C. Here PRMν means a Poisson random measure with points in C whose mean measure is ν. The two convergence statements in 3.2 mean that extremes of each sample when taken in C will asymptotically have the same properties. See Resnick Mixture characterization when the hidden angular measure is finite. We now show that a regularly varying distribution on E possessing hidden regular variation on E is tail equivalent on E and E to a mixture model, provided the hidden angular measure is finite. The following definitions facilitate our presentation. Suppose Y is a random vector whose distribution is multivariate regularly varying on C and satisfies 2.. Call Y extremally dependent on C, if the limit measure ν is non-zero on C. If C, C = E, E, then this means that Y is not asymptotically independent. A multivariate regularly varying random vector Y is called completely asymptotically independent, if its support is bounded away from. Then for some t >, the support is contained in t, c that is d i= Y i < t a.e. and hence cannot have any hidden regular variation, though it will surely be asymptotically independent. Therefore, an exactly independent random vector, that is, a vector with independent components, cannot be completely asymptotic independent cf. Resnick, 22b, Example 2. A completely asymptotically independent vector is tail equivalent on the cone E to a random variable Y which is a mixture of regularly varying distributions concentrating on the axes. This means Y can be represented as follows: Let I, {X i, i =,..., d} be independent, I takes values in {, 2,..., d} with equal probabilities and {X i } i.i.d. with one-dimensional distribution F x with F x RV α. Then Y is tail equivalent on E to d j= I=jX j e j. Also note that the parameters χ, and χ cf. Coles et al., 999, Heffernan, 2 will have values and respectively. The two concepts of extremal dependence and complete asymptotic independence represent two extreme situations of hidden regular variation. 4.. Characterization for finite hidden angular measure. The next Theorem summarizes how tail behavior of the distributions on E possessing hidden regular variation on E with finite hidden angular measure can be characterized in terms of tail equivalence. Theorem 4.. Suppose F is a multivariate regularly varying distribution on E with hidden regular variation on E, finite hidden angular measure, and scaling functions bt, b t with bt/b t. Then F is tail equivalent on both the cones E and E to a distribution which is mixture of i a completely asymptotically independent distribution on E with no hidden regular variation whose marginal distributions have scaling function bt, and ii a distribution on E which is extremally dependent on E with scaling function b t. The tails of this extremally dependent distribution are thus lighter than those of the completely asymptotically independent distribution. i=

6 6 K. MAULIK AND S. RESNICK Conversely, if F is tail equivalent to a mixture as above with bt/b t, then F is multivariate regularly varying distribution on E and has hidden regular variation on E with finite hidden angular measure and with scaling functions b and b. Proof. Let Y be a d-dimensional random vector distributed as F, which is multivariate regularly varying on E with hidden regular variation on E. Suppose Y has finite hidden angular measure which, without loss of generality, we take to be a probability measure. Then Y satisfies the conditions 2.3 and 2.5 with c =. Extend the hidden angular measure S, defined in 2.5, to ℵ by defining it to be on ℵ \ ℵ. Recall that the one-dimensional marginal distributions of Y are identical. Take i.i.d. random variables X,..., X d having the same distribution as Y and independent of the independent pair R and Θ all defined on the same probability space, where Θ is distributed as S and R is distributed as F given by {, x 4. F x =, x, x b since b =. We further take an integer valued random variable I independent of all the rest and defined on the same probability space as before, which takes value with probability 2 and values, 2,..., d with probability 2d each. Define V to be the random vector by transforming the pair R, Θ to Cartesian coordinates, that is, V = R Θ. Finally define d 4.2 Z = {I=} V + {I=i} X i e i, and we claim that the distribution of Z is tail equivalent to F on both E and E. First we check that V is multivariate regularly varying on E with scaling function b t and hence also on E. We check this using polar coordinates through the convergence in 2.3. For any Borel set Λ ℵ and x >, RV 4.3 b t > x, Θ R V Λ = b t > x S Λ = t b b tx S Λ x α S Λ. Also since S, S ℵ \ ℵ =, V is extremally dependent. So Z is a mixture of a multivariate regularly varying random vector V which is extremally dependent and a random vector whose distribution is multivariate regularly varying and completely asymptotically independent. Next observe that d 4.4 Θ Z = {I=} Θ + {I=i} e i and 4.5 i= R Z = {I=} R + i= d {I=i} X i. Then for Λ, a compact subset of ℵ and hence e i / Λ, i =,..., d, we have that RZ b t > x, Θ R d Z Λ = I =, b t > x, Θ X i Λ + I = i, b t > x, e i Λ i= = 2 R > b txs Λ + i= x α S Λ, by 4.3, since e i / Λ, for i =, 2,..., d. Therefore, RZ b t, Θ v Z 2 ν α S on, ℵ.

7 Again, for Λ, a subset of ℵ, we have that RZ bt > x, Θ Z Λ = I =, R bt > x, Θ Λ 4.7 HIDDEN RULAR VARIATION 7 + d i= = 2 R > btxs Λ + 2 X > btx d + c 2 x α d d {ei}λ, i= I = i, X i bt > x, e i Λ d {ei}λ where c = ν,, d. The in the last line of the previous display results from b bt since bt/b t implies b t/b t which in turn implies /b bs as s. Thus RZ bt, Θ v Z c 2 ν α S on, ℵ, where S is the uniform distribution on {e,..., e d }. Therefore, Y and Z are asymptotically equivalent on both the cones E and E. Conversely, suppose Y has hidden regular variation and i and ii hold with bt/b t. Since a completely asymptotically independent distribution cannot have a non-zero hidden angular measure, and the scaling function b t of the extremally dependent part is of smaller order than that of the completely asymptotically independent part, the hidden angular measure of F is a multiple of the hidden angular measure of the extremally dependent part, which is finite Significance of a finite angular measure on E. Before considering distributions with infinite hidden angular measure, we highlight the significance of having a finite angular measure. Having a finite hidden angular measure on E means that regular variation on E can be extended to regular variation of the same order on the full cone E and that marginals are regularly varying. Theorem 4.2. Suppose V is regularly varying on E, with index α, scaling function b t, limit measure ν, angular measure S on ℵ. The following are equivalent: i S is finite on ℵ. ii There exists a random vector V defined on E such that and for i =,..., d V tee V V i > b tx cx α, t for some c >, so that each component V i has regularly varying tail probabilities with index α. iii There exists a random vector V defined on E such that tee V and V is regularly varying on the full cone E with scaling function b t and limit measure ν and V ν E = ν ; that is, the restriction of ν to E is ν. iv There exists a random vector V defined on E such that tee V such that for any s, \ {} and any a, \ d i= {te i : < t < } d d i= s i V i and V i= a i V i, i=

8 8 K. MAULIK AND S. RESNICK are tail equivalent on, and have regularly varying tail probabilities of index α. Here we define e i =,...,,,,..., for the vector whose ith component is and whose other components are. Remark 4.. We cannot replace V with V and still hope the assertions to hold. In 4.2, we have Z regularly varying on E but the marginal distributions are not regularly varying with index α. Proof of Theorem 4.2. i iii: S is defined on E. It can be extended to all of E by setting S e i =. Now repeat the construction of R, Θ given prior to and including 4. and this time define V = R Θ on E which will be regularly varying on E with scaling function b t, limit measure ν with ν E = ν. iii i: If V is regularly varying on E, then Therefore iii ii: Since on E, we have that V b t > x V > b t ν{x E : x > } <. ν{x E : x > } = ν {x E : x > } = S ℵ <. V b t ν V = x,, d ν x,, d. b t This implies marginal regular variation. ii iii: Suppose V is regularly varying on E and the one dimensional marginal tails are also regularly varying. For x E such that at least 2 components are strictly positive, we have by inclusion-exclusion c V b t x = =t d i= d P i= V i b t > xi V i b t > xi V i b t > xi, V j b t > xj + i<j d V + d+ b t > x. Convergence of the first sum results from one-dimensional marginal convergence. Convergence of the other terms results from vague convergence on E. To understand iv, see Theorem of Resnick 22b. 5. Hidden regular variation when the hidden angular measure is infinite. Techniques used in the previous section when the hidden angular measure was assumed finite need modification when the hidden angular measure is infinite. We discuss the differences between the two cases, provide tools for discussing regular variation on a subcone of E, give a broad class of examples for this infinite case and provide a characterization. 5.. Regular variation on E. When the hidden angular measure was finite, regular variation on E could be extended to all of E with the same index; see Theorem 4.2. The situation when the hidden angular measure is infinite, is rather different. If regular variation can be extended to all of E, then the order of the extension is different, in the sense that we need a scaling function on E which is asymptotically heavier than what is needed on E.

9 HIDDEN RULAR VARIATION 9 Theorem 5.. Suppose V is defined on E, has equal marginal distributions, and is regularly varying on E, with index α, scaling function b t RV /α, limit measure ν, infinite angular measure S on ℵ. Then V is also regularly varying on E iff there exists α α such that P V i > x RV α, i =,..., d, and with the quantile function 5. bt = t PV i > satisfying bt/b t as t. Proof. Only if part: Suppose V is regularly varying on E with limit measure ν. We may assume the quantile function given in 5. is the scaling function of the regular variation. If νe >, then for some x >, V b t > x ν x, >, V bt > x νx, > which implies bt cb t, for some c >. This means that regular variation on E is extended to all of E and, by Theorem 4.2, S, the hidden angular measure, is finite. This is a contradiction and hence we conclude we must have νe =. But then we have PV /bt > x PV /b t > x, which implies bt/b t as t. Thus we conclude α α. Also V V bt x = x,, d ν x,, d bt which implies PV > x RV α. If part: The converse follows from Corollary in Resnick 22b without the assumption of infinite hidden angular measure A coordinate system for regular variation on a subcone; the inverse norm transformation. To study the behavior of the random vector around, it is useful to study the coordinate by coordinate reciprocal of the vector around. This motivates us to consider the map x x from,, ; that is x,..., x d x,...,. x d Here we define / = and / =. Let be a norm on R d. Extend the definition from, to, by defining x = if any component x i of x is infinite. Now define and inv :,,, by x inv = x, ℵ inv = {x E : x inv = }. Note x inv = iff x = and x inv = iff d i= xi =. Furthermore, observe that inv is homogeneous: tx inv = t x inv and that {x E : x inv } is a compact neighborhood of. If x = d i= xi, then d x inv = d i= xi = i= xi d i= j i. xj

10 K. MAULIK AND S. RESNICK If x = d i= xi then x inv = d i= xi. In this case, the set {x E : x inv } is the compact hypercube, whose diagonally opposite vertices are and. This has the advantage of being immune to whether the hidden angular measure is infinite or not. Thus, if ν is the limit measure in the definition of regular variation on E, } x S inv Λ := ν {x E : x inv > ; Λ x inv is a finite measure on ℵ inv. Because of homogeneity of both inv and ν we also have the product form: } x ν {x E : x inv > r; Λ = r α S inv Λ x inv for r > and Λ a Borel subset of ℵ inv. Note for d = 2, S inv ℵ inv > but for d > 2, it is possible that S inv. See Example 5.2 below. Recall from Section.3, that the domain of the polar coordinate transform is, \ {}. Define T inv :, \ { }, ℵ inv ; T inv x = x inv, x x inv so that T inv :, ℵ inv, \ { }. For a random vector V defined on, define T inv V = R inv, Θ inv. Regular variation on the subcone, can be characterized in terms of these coordinates. Theorem 5.2. For a random vector V on E, V is regularly varying on E =, with scaling function b t, limit measure ν : V b t v ν, in M + E iff Rinv b t, Θ inv in M +, ℵ inv, with c,. v c ν α S inv The proof is similar to the one used to express multivariate regular variation on E in terms of the usual polar coordinates and is hence omitted. Remark 5.. The characterization in Theorem of Resnick 22b shows why in Theorem 5.2 we cannot get a characterization in terms of transformed coordinates on E. Note, however, when d = 2 that E = E. The following example gives further insight into why a characterization on E is not possible. Example 5.. Suppose d = 3 and X, X 2, X 3 are i.i.d. Pareto random variables with parameter. Then one checks that α =, α = 2 and α = 3. Thus the regular variation behavior on E is essentially different from the regular variation on E, as the latter is determined by the behavior on the two-dimensional faces. Except for some clearly noted exceptions, we proceed assuming d = 2 and assuming x inv = x x Hidden regular variation in standard form. Theoretical developments are often worked out for multivariate regular variation when a standard form is assumed cf. Resnick, 987, pp. 265, 277 which allows the scaling function to be linear. For hidden regular variation, there are two scaling functions and we have to choose which one to make linear. It is more convenient to transform so that the hidden scaling function b t becomes linear. Theorem 5.3. Suppose d = 2. The vector V is regularly varying on E and E with limit measures ν, ν, indices α, α, and scaling functions bt, b t = / P V inv > t with bt/b t iff for 5.2 Ux = P V inv > x, we have UV := UV, UV 2 regularly varying on E, E with limit measures ν, ν, indices α/α, and scaling functions b t = U bt and b t = t.,

11 Proof. For regular variation on E : We have and therefore for x >, UV t HIDDEN RULAR VARIATION U b t t, > x = V > U tx, U tx 2 V U = U t > tx U, U tx 2 t U t ν x /α, x 2 /α, =: ν x,. For regular variation on E: We have UV i lim t Ubt > x V i = lim t bt > U xubt bt V i = lim t bt > U xubt U Ubt since U Ut t and because Ubt, we get this limit to be U α tx = lim t U = x α/α, t where α/α. Take the inverse norm transform of the coordinates in standard form. Define R = UV inv = UV UV 2 ; Θ = UV R. Corollary 5.. Suppose d = 2 and the vector V possesses hidden regular variation and is regularly varying on E and E with limit measures ν, ν, indices α, α, and scaling functions bt, b t = / P V inv > t with bt/b t. In standard form, for R, Θ we have R t, Θ dr dθ v r 2 dr S inv dθ in M +, ℵ inv, where S inv is a finite measure on ℵ inv. If U given in 5.2 is continuous and strictly increasing, R is Pareto with parameter. Remark 5.2. Note that S inv ℵ inv is given by S inv ℵ inv = lim t R > t. Since the set R > t = UV > t, UV 2 > t is compact in E, the limit will be finite. Proof. This follows from the fact that U is non-decreasing and 2 2 U V i = U V i. i= i=

12 2 K. MAULIK AND S. RESNICK Identifying an infinite hidden angular measure. We can identify when the hidden angular measure is infinite using S inv. Proposition 5.. Suppose d = 2 and define the finite measures G i, i =, 2 by Ḡ s = S inv {} s,, Ḡ2 s = S inv s, {}, s. The limit measure for regular variation on E has infinite angular measure, or equivalently, iff 5.3 max i=,2 ν {x E : x > } =, Ḡ i sds =. Without standardization, the condition corresponding to 5.3 becomes max i=,2 s α Ḡ i sds =, where Ḡs = ν { x, x 2 : x, x /x 2 > s } and Ḡ2s is defined similarly. Proof. Let < x < x 2. We have ν x, =ν {y E : y > x} y = ν {y : y inv > x} y inv = ν dr dθ = r 2 drs inv dθ r, θ ℵ inv rθ>x = = θ ℵ inv θ ℵ inv r> x x2 θ θ 2 x x2 θ θ 2 and splitting ℵ inv = {},, {} we get and thus = 5.4 ν x, = x 2 r, θ ℵ inv rθ>x r 2 drs inv dθ S inv dθ x s G x 2 ds + x 2 /x sg ds + G 2, s x G x 2 2 ds + x Ḡ x 2 /x. Let x. Then ν, x 2, = iff Ḡ sds =. Similarly, if we interchange the roles of x and x 2, we get that ν x,, = iff Ḡ 2 sds =. If we suppose S inv is a probability measure on ℵ inv and that Θ inv is a random vector on ℵ inv with distribution S inv, then the previous result requires max E Θ inv i =. i=,2 Though the above proposition has been given only for the case d = 2, it can be easily extended to d > 2. The extension follows from the following lemma, where we use the max-norm, namely x = d i= xi :

13 HIDDEN RULAR VARIATION 3 Lemma 5.. Suppose X is a random vector of dimension d > 2, which is multivariate regularly varying on E with hidden regular variation on E, having limit measures ν, ν. Define Ẽ and Ẽ to be the spaces corresponding to E and E in two dimensions; that is Then for some pair i < j d, necessarily Ẽ =, 2 \ {, }; Ẽ =, ν {x E : x i >, x j > } and for all such pairs we must have X i, X j to be multivariate regularly varying on Ẽ with hidden regular variation on Ẽ. Also the hidden angular measure of X is finite iff for all pairs i < j d, satisfying the condition 5.5, the hidden angular measure of X i, X j is finite. Remark 5.3. Condition 5.5 is automatic when d = 2 but cannot be dropped for d > 2. The identical marginal distributions do not impose any restriction on the two-dimensional behavior as is illustrated in the following example. There is no hidden regular variation for X, X 2. Example 5.2. Suppose U i, i =, 2, 3 are i.i.d. Pareto random variables on, with parameter. Let B i, i =, 2 be i.i.d. Bernoulli random variables with and {B i } and {U i } are independent. Define Then P B i = = P B i = = 2, W = B U 3, X = B 2 U,, W + B 2, U 2, W. X = B 2 U, X 2 = B 2 U 2, X 3 = W are identically distributed, having atom sized 2 at and having a Pareto density with parameter and total mass 2 on,. The distribution of X is supported on the planes where either of the first two coordinates vanish and furthermore X inv =. Thus S inv. It is easy to see that X is multivariate regularly varying on E with α = and bt = t and has hidden regular variation on E with α = 2 and b t = t. However, ν {x : x >, x 2 > } =. So hidden regular variation is not possible for the marginal distribution of X, X 2. Also, this is an example, where we have hidden regular variation on E, but not E. This emphasizes the lesson learned in Example 5., that regular variation on E and E can be quite different. Proof of Lemma 5.. First observe that E = i<j d {x : x i >, x j > } and since ν is a non-zero measure on E, the condition 5.5 holds for some pair i < j d. As usual, let b and b be the scaling functions for X with indices /α and /α respectively. Now, for x = x, x 2, the d-dimensional set A ij = {y : y i > x, y j > x 2 } is relatively compact in E and hence X i b t > x, Xj X b t > x2 = b t A ij ν A ij =: ν,i,j x, x 2,. Note that whenever the pair i, j satisfies the condition 5.5, the limiting measure ν,i,j is non-zero. Also, for any i =,..., d X i bt > x x α. So if i, j satisfies the condition 5.5, X i, X j is multivariate regularly varying on Ẽ and has hidden regular variation on Ẽ. The scaling functions and indices remain the same. Also if the pair i, j does not satisfy the condition 5.5, then clearly ν,i,j is identically the zero measure. Now for a pair i, j satisfying the condition 5.5, the hidden angular measure of X i, X j is finite iff ν,i,j {x, x 2 Ẽ : x x 2 > } = ν {x E : x i x j > } <.

14 4 K. MAULIK AND S. RESNICK Suppose X has finite hidden angular measure. Then d >ν {x E : d i=x i > } = ν {x E : x i > } i= ν {x E : x i } {x E : x j } = ν,i,j {x, x 2 : x x 2 }. Conversely, if for all i < j d satisfying 5.5, the vector X i, X j has finite hidden angular measures, then ν {x E : d i=x i > } =ν x E : x i x j > i<j d ν {x E : x i x j > } i j = ν,i,j {x, x 2 : x x 2 > } <, i<j d since we are summing a finite number of finite terms and zeros. Combining Proposition 5. and Lemma 5., we have the following result: Theorem 5.4. Suppose X is a d-dimensional random vector defined on, which is multivariate regularly varying on E and has hidden regular variation on E with limit measure ν. Then ν has finite angular measure iff for all pairs i, j, i j the function { } Ḡ i,j s := ν x E : x, xi x > s j satisfies s α Ḡ i,j sds < A partial converse to a theorem of Breiman. A theorem of Breiman 965 discusses the tail behavior of a product of two independent, non-negative random variables, one of which has regularly varying tail probabilities and the other has a lighter tail. Our class of examples in the next section requires the following partial converse. Proposition 5.2. Suppose ξ and η are two independent, non-negative random variables and ξ has a Pareto distribution with parameter : Pξ > x = x, x. iff a We have and then Pξη > x RV α, α <, Pη > x RV α, Pξη > x Pη > x α. b If Pξη > x RV and ξη has a heavier tail than ξ meaning Pξη > x Pξ > x = x Pη > ydy,

15 i.e., Eη =, then HIDDEN RULAR VARIATION 5 x Pη > sds =: Lx is slowly varying. If in addition Lx Π, the de Haan function class Π cf. Bingham et al., 987, de Haan, 97, Geluk and de Haan, 987, Resnick, 987, then and Pη > x RV Lx Pξη > x =. x Pη > x Pη > x Remark 5.4. The assumption that the integral is Π-varying in b is implied by supposing thaη > x RV. We see from b, that the product has a heavier tail than either factor random variable. As an example, consider Pη > x = e log x, x > e. x An easy calculation shows that Pξη > x 2 ex log x 2, x. Proof of Proposition 5.2. a If part: If Pξη > x RV α, the result follows from Breiman 965 or Embrechts and Goldie 98. Only if part: We have Pξη > x = P η > x x s 2 ds = x Pη > ydy. s So Pξη > x RV α, α < iff x Pη > ydy RV α. By the monotone density theorem and Karamata s theorem Bingham et al., 987, Geluk and de Haan, 987, Resnick, 987, we have Pη > y RV α and then by Karamata s theorem Pξη > x Pη > x = x Pη > ydy x Pη > x α. b This follows from the fact that the derivative of a Π-varying function U with a non-increasing derivative U is -varying with auxiliary function xu x and that Ux/xU x. See Bingham et al. 987, de Haan 976, Geluk and de Haan 987, Resnick 987. Remark 5.5. Assuming one factor random variable is Pareto, this result shows the product random variable has a heavier distribution tail which is regularly varying iff the other factor random variable has a tail which is regularly varying. The result fails without the assumption that one factor is Pareto. The following very clever example devised by Daren Cline is an example where ξη has a Pareto tail, the tail of ξη is heavier than that of ξ, but η does not have a regularly varying tail. Example 5.3 Cline. Let F be any distribution on, with finite first moment and set F 2 x = F e π x. Define C i = Then C 2 = e π C and S 2 = e π S. Define so that x coslog xf i dx and S i = F x = eπ F x + F 2 x e π, + x coslog xf dx = x sinlog xf i dx. x sinlog xf dx =.

16 6 K. MAULIK AND S. RESNICK Now let for y >. Note that its derivative is Ḡy = +.5 sinlog yy, +.5 sinlog y +.5 coslog y y 2 <. Suppose ξ has distribution F and η has distribution G. Then Pξη > t = Ḡt/xF dx = =t xf dx +.5 sinlog t.5 coslog t =t xf dx. +.5 sinlogt/xt/x F dx x coslog xf dx x sinlog xf dx Thus, Pξη > t is -varying, Ḡ is not -varying and F t = opξη > t A class of examples of hidden regular variation with infinite hidden angular measure. Let d = 2 and suppose R eg is a Pareto random variable on, with parameter and Θ eg is a random vector living on ℵ inv and for some probability distribution G concentrating on,, we have 5.6 PΘ eg {} s, = PΘ eg s, {} = s. 2Ḡs, Suppose R eg and Θ eg are independent. Define 5.7 V eg = R eg Θ eg. The assumption 5.6 is the same as assuming that 5.8 V d = V 2 which is consistent with our standing assumption that marginal distributions are equal. Observe that by 5.6, we have for s >, PΘ > s = PΘ eg s, {} = 2Ḡs = PΘ eg {} s, = PΘ 2 > s, so that 5.6 implies 5.8. Conversely, if 5.8 holds, then R eg Θ = R eg Θ 2 and therefore P Θ > x y 2 dy = P Θ 2 y > x y 2 dy. y This is equivalent to and it follows that x P Θ > s ds = x d P Θ 2 > s ds P Θ > s = P Θ 2 > s, s >. Proposition 5.3. With V eg as specified, V eg is regularly varying on E with index α =. The limit measure ν eg, has infinite angular measure iff 5.9 Furthermore, if 5.9 holds: Ḡsds =.

17 HIDDEN RULAR VARIATION 7 i V eg is regularly varying on E with index α < and hence possesses hidden regular variation with infinite hidden angular measure iff G RV α. In this case, for i =, 2, we have P V i > x x. 2 αḡx, ii V eg is regularly varying on E with index α = and possesses hidden regular variation with infinite hidden angular measure iff Lx := x Ḡsds RV and Lx. A sufficient condition is Ḡ RV with Ḡsds = in which case P V i > x P V i lim = lim > x =, x P R eg > x x P Θ i > x for i =, 2. Proof. Note, that since Θ eg lives on ℵ inv, we have Θ eg inv = and thus R inv = R eg. Similarly, since we have Hence, for this example, V eg inv V eg, t V eg inv R eg Θ eg inv = R eg Θ eg inv V eg V eg inv = Θ eg. Reg dr dθ = dr PΘ eg dθ r 2 dr PΘ eg dθ. t Regular variation on E follows from Theorem 5.2. The statement about when the angular measure is infinite follows from Proposition 5.. The statement about regular variation on E comes from Proposition 5.2 and Theorem 5.. The asymptotic form for the marginal tail of V eg comes from the fact that for x >, 5. P V i > x = P R eg Θ i > x = =x x = 2x + PΘ i x > ydy Ḡydy and the result follows from application of Karamata s theorem. P Θ i > x r r 2 dr This result shows that the class of multivariate distributions which are regularly varying on E with infinite angular measure is at least as large as the class of distributions on, which have infinite mean. The class of multivariate distributions on E which are regularly varying and possess hidden regular variation is at least as large as the class of distributions on, which have regularly varying tails of index less than. For this class of examples, we have V eg = R eg Θ eg, and thus V V 2 = Θ Θ 2.

18 8 K. MAULIK AND S. RESNICK One way to contruct Θ eg is as follows. Let Θ be a random variable concentrating on, with distribution G and suppose B is a Bernoulli random variable with equally likely values and, independent of Θ and R eg and then set Θ eg = BΘ, + B, Θ, so that Θ eg concentrates on ℵ inv and has distribution given in 5.6. Then Θ = BΘ + B Θ 2 Θ. Since /Θ it does not affect tail behavior in the mixture and we can write, as x, Θ Θ 2 P > x = P > x = Θ 2 Θ 2 PΘ > x + 2 PΘ > x PΘ > x = 2 2Ḡx. We conclude that for this class of examples, V eg is regularly varying on both E and E with indices α,, with α <, iff V RV V 2 α. However, more general cases of regular variation on E and E will not necessarily follow this pattern. Consider the following example. Example 5.4. Suppose β < α < and that U i, i =, 2 are i.i.d. random variables concentrating on, having Pareto distributions with parameter β. Let M be a multinomial random variable with values {, 2, 3} and equal probabilities /3. Suppose M is independent of U i, i =, 2 and V eg is constructed as above. Define X = M= V eg + M=2, + M=3,. U 2 Since /U i, these terms cannot affect tail behavior: X > x t c X tp bt x U 3 V eg t 3 tp V eg bt x > x, x >, c, where bt is the scaling function of V eg for the cone E. Thus X is regularly varying on E and E. However, so that X X 2 = M= V V 2 + M=2 U + M=3 U 2 X P X > x RV 2 β. Example 5.5. Suppose Z = Z, Z 2 are i.i.d. Pareto random variables on, with parameter /2. Then Z t v ν in E where which has density ν x, =, x >, x x 2 ν x = 4 x x 2 3/2.

19 An easy calculation shows that HIDDEN RULAR VARIATION 9 { } ν x : x ; x x > θ = θ /2 const, 2 which is consistent with the class of examples discussed. The following gives conditions for a distribution having hidden regular variation to be tail equivalent on E and E to V eg. Proposition 5.4. Suppose X has a distribution possessing hidden regular variation in standard form on E and E, with limit measures ν, ν with ν {x : x > } =, scaling functions bt RV /α, α <, b t = t, and such that S inv ℵ inv = ν {x E : x, x inv = } =. Define and V eg as in 5.7. Then iff the following statements hold: i We have, for θ >, and ii we have Ḡt = S inv {} t, = Sinv t, {}, X tee,e lim X > t, X2 t lim t V eg X 2 > t, X X 2 > θ X > θ = 2Ḡθ, = 2Ḡθ, Ḡ RV α, and PXi > x c,, x. Ḡx If α =, bt/t as t, the result continues to hold provided ii is replaced by ii We have for some c >. x Ḡsds RV and x Ḡsds cx PX i > x, x Proof. Suppose X has a distribution which is regularly varying on E, E with parameters α, and which is tail equivalent to V eg on E, E. Then from tail equivalence on E, for i =, 2: PX i > t c PV i > t = c PR egθ i > t which implies PΘ i > t RV α, by Proposition 5.2. Therefore, Ḡt RV α which gives ii. Furthermore, tail equivalence on E gives, for θ >, lim t X > t, X2 X > θ = lim t X X 2 > t, X {} θ, X X 2 = lim R eg > t PΘ eg {} θ, = t 2Ḡθ, which gives i. Conversely, suppose X has a distribution which is regularly varying on E, E and i and ii hold. Condition i implies X tee V eg. Condition ii guarantees in light of Theorem 5. tail equivalence on E.

20 2 K. MAULIK AND S. RESNICK When α =, we use 5., which says P V i eg > x = + 2x x Ḡydy. Also note that the hidden regular variation of X with α = and b t = t implies x P X i > x is a slowly varying function going to. Referring to Example 5.5 we observe for θ >, Z > t, Z2 Z > θ =t t =θ /2 t 2 2 s 3/2 θs /2 ds t s 2 ds = θ /2 2 = Ḡθ. Full disclosure: While this result clarifies when a distribution possessing hidden regular variation with infinite angular measure has the structure of V eg, it does not provide a full characterization Random vectors with prescribed infinite hidden angular measure. The examples of random vectors with infinite hidden angular measure given in the earlier subsections are of dimensions 2 and 3. The construction given in Subsection 5.5 is for dimension 2. In this subsection, we again consider general dimensions. Given an infinite hidden angular measure S and scaling function b satisfying bt/t, we construct a random vector Z in standard form with scaling function b on E and hidden angular measure S. We use the standard form for notational simplicity. The case of a general scaling function b for the subcone E can be easily obtained through the transformation outlined in Theorem 5.3. As in Theorem 4., define the random variables I, X,..., X d. Each X i has the property that for x >, tp X i > btx x α. Now we specify the random vector V which will give regular variation on E. Define the non-decreasing, left continuous function { Φt = S a ℵ : d i= a e i > }. t Since the hidden angular measure S is infinite, Φ increases to. Also define a non-decreasing function φ as follows: { } φt = inf log b2u 2 u : u t. Since bt/t, φ also increases to. Further observe that { } { φlog 2 t = inf log b2u 2 u : u log 2 t = inf log bu } u : u t log bt t and hence we have which further implies φlog lim 2 tt =, t bt b φlog 5. lim 2 tt =, t t since b is regularly varying of index α >. Now we define an increasing sequence of pre-compact open covers {G n } n= for ℵ, none of which has too large S -mass: G = and for n, { } G n = a ℵ : d i= a e i >. Φ φn Thus 5.2 S G n \ G n S G n = Φ Φ φn φn,

21 since Φ is left continuous. Define the function fa = 2 n φn Gn\G n a HIDDEN RULAR VARIATION 2 n= on ℵ. Then f is bounded on compact subsets of ℵ and also satisfies 5.3 c := fa S da fa S da = 2 n S G n \ G n 2 n = <. ℵ ℵ φn Now define a pair of random variables R, Θ, taking values in, ℵ, independent of I, X,..., X d, with joint distribution c S dθf dr {r,θ :r fθ }r, θ, where F is as defined in 4., with b as the identity function. Hence F is nothing buareto distribution function with parameter. The joint distribution is a probability distribution since S dθf dr = S dθf fθ = fθ S dθ =, c {r,θ, ℵ :r>fθ} c ℵ c ℵ by 5.3. Define V = R Θ. Finally define Z as in 4.2 by d Z = I= V + I=i X i e i. This is the required random vector Z. To verify the hidden regular variation of Z on the cone E, first observe that given any compact subset Λ of ℵ and x >, we may choose N such that Λ G N. Then for all t > 2 N φn/x and θ Λ, we have, fθ < tx and hence 5.4 R > tx, Θ Λ = S ΛtF tx x S Λ, which is the analog of 4.3 in the finite hidden angular measure case with α =. The decompositions 4.5 and 4.4 of R Z and Θ Z respectively continue to hold. So we get the analogs of 4.6 in the infinite case as well. Verifying regular variation on the bigger cone E is more difficult. We first check that scaling by b is too large for V on the cone E by checking that for any x >, It suffices to suppose x =. First define, for t >, Then κ is increasing to. Also we have i= n= R > btx. κt = sup {u : 2 u φu bt}. 5.5 φκt2 κt bt. Furthermore, we have, and hence 5.6 Then observe that, and φκt + 2 κt+ > bt b φκt + 2 κt+ > t fθ φ κt 2 κt bt, for θ G κt, fθ φ κt + 2 κt + > bt, for θ / G κt. n=

22 22 K. MAULIK AND S. RESNICK Then we have cr > bt = c R > bt, Θ G κt + c R > bt, Θ / G κt = t F bts dθ + t F fθs dθ G κt G c κt = ts G κt + t fθ S dθ bt G c κt tφ κt + t 2 n S G n \ G n, by 5.2 bt φn tφκt bt n= κt + 3t2 κt, by t2 κt, since φ is increasing and by 5.2 < 6 b φκt + 2 κt+ 2 κt+, by 5.6 = 6 b φlog 2 uu, substituting u = 2 κt+. u Now, using 5., we have this quantity going to zero, as required. Then, again as in Theorem 4., Z is regularly varying on E. So Z is a random vector having multivariate regular variation on E with scaling function b and having hidden regular variation on E with scaling by the identity function and having prescribed infinite hidden angular measure S. Thus, any random vector Y which is regularly varying on E and E with scaling functions b, b and infinite hidden angular measure S is tail equivalent to a mixture I= V + I X where X is completely asymptotically independent and V is regularly varying on E with tp V > btx, x >. 6. Can hidden regular variation influence the tail behavior of the product of the components? The tail behavior of distributions of products of heavy tailed variables arises fairly frequently. For the study of the heavy tailed sample correlation function, the tail behavior of X X 2 where X, X 2 are iid with regularly varying tails of index α was crucial and found to partly depend on whether EX α was finite or infinite. See Cline 983, Davis and Resnick 985, 986. In Internet studies of file downloads, download throughput or rate R and download duration L are heavy tailed and determine the size of the file F since F = RL Maulik et al., 22. Hidden regular variation condition on the cone C = E may or may not influence the tail behavior of the distribution of the product of the components. For dimensions 3 or higher, problems may be more pathological due to the flexibility in the choice of the subcone C. See Example 5.2. The hidden regular variation definition does not prevent the limiting measure ν from concentrating on the hyperfaces of the d-dimensional non-negative orthant. On the hyperfaces, at least one of the coordinates will be zero and so will be the product. This problem suggests we examine the convergence more carefully and seek the right scaling to understand the behavior on the subcone E =, =, instead of E. However, the problem is deeper than the choice of the correct subcone. In case d = 2, the cone, coincides with E. As the following counterexample shows, even in that case hidden regular variation is not enough to characterize the behavior of the product of the components.