Extremal dependence of random scale constructions

A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction $(X_1,X_2)=R(W_1,W_2)$, with non-degenerate $R>0$ independent of $(W_1,W_2)$. Focusing on the presence and strength of asymptotic tail dependence as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of $R$ and $(W_1,W_2)$, the shape of the support of $(W_1,W_2)$, and dependence between $(W_1,W_2)$. When $R$ is distinctly lighter tailed than $(W_1,W_2)$, the extremal dependence of $(X_1,X_2)$ is typically the same as that of $(W_1,W_2)$, whereas similar or heavier tails for $R$ compared to $(W_1,W_2)$ typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic dependence and asymptotic independence of $(X_1,X_2)$ possible in such cases when $(W_1,W_2)$ exhibit asymptotic independence. Our results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.


Introduction
A rich variety of bivariate dependence models have a pseudo-polar representation (X 1 , X 2 ) = R(W 1 , W 2 ), R > 0, independent of (W 1 , W 2 ) ∈ W ⊆ R 2 , where we term R the radial variable, assumed to have a non-degenerate distribution, and (W 1 , W 2 ) the angular variables. Indeed, many well-known copula families, including the elliptical, Archimedean, Liouville, and multivariate Pareto families have such a representation. In this work, our focus is on the upper tail dependence of such constructions. In particular, we examine whether a given (X 1 , X 2 ) displays asymptotic dependence or asymptotic independence, and the strength of dependence within these classes. Our results are particularly useful for constructing new models with properties that reflect the challenges of real data in, for instance, finance, meteorology and hydrology. Specifically, it is often ambiguous whether data should be modeled using an asymptotically dependent or asymptotically independent distribution, and most families of distributions only exhibit one type of dependence. Through our results we are able to present new models that smoothly bridge these two classes. A vector (X 1 , X 2 ) with X j ∼ F Xj is said to display asymptotic dependence if the limit exists and is positive; a limit of zero defines asymptotic independence. In (2) and throughout, F −1 Xj denotes the (generalized) inverse of the distribution function F Xj . The parameter χ X is termed the (upper) tail dependence coefficient, and the value of χ X ∈ (0, 1] summarizes the strength of the dependence within the class of asymptotically dependent variables. Under asymptotic independence, a more useful summary is the rate at which the convergence to zero in equation (2) occurs, and a widely satisfied assumption Tawn, 1996, 1997) is where : [0, 1] → R + is slowly varying at zero, i.e., lim s→0 (sx)/ (s) = 1, x > 0. The parameter η X is termed the residual tail dependence coefficient; positive and negative extremal association are indicated respectively by η X ∈ (1/2, 1] and η X ∈ [0, 1/2), whilst asymptotically dependent variables have η X = 1 and χ X = lim q→1 (1 − q). A value of η X = 0 means that the left-hand side of (3) decays faster than any power of 1 − q, whilst if the left-hand side is exactly zero for some q < 1, we say that η X is not defined. Our particular interest in the extremal dependence of constructions of the form (1) stems not so much from their novelty, but from their ubiquity and flexibility. As mentioned, (1) already encompasses many well-known families, and moreover these families display different types of extremal dependence, which may be determined by the distribution of R, the distribution of (W 1 , W 2 ), or its support W. There is a large body of literature that treats either individual constructions of the form (1), or a particular subset of these constructions where R or (W 1 , W 2 ) have certain specified properties; this literature will be reviewed in Section 4. Our aim is to bring this scattered treatment together and more systematically characterize how the extremal dependence of (X 1 , X 2 ) is determined by the properties of R and (W 1 , W 2 ). By understanding which facets of the construction lead to different dependence properties, we are able to determine dependence models that can capture both types of extremal dependence within a single parametric family; the recent proposals in Wadsworth et al. (2017), Huser et al. (2017) and Huser and Wadsworth (2018) are specific examples of this.
A broad split in representations of type (1) is the dimension of W, the support of (W 1 , W 2 ). The most common case in the literature is that W is a one-dimensional subset of R 2 , such as the unit sphere defined by some norm or other homogeneous function. Examples include the Mahalanobis norm (elliptical distributions), L 1 norm (Archimedean and Liouville distributions), or L ∞ norm (multivariate Pareto distributions). On top of the support W, to obtain distributions within a particular family, R or (W 1 , W 2 ) may be specified to have a certain distribution. Where W is two-dimensional, it may sometimes be reduced to the one-dimensional case by redefining R, such as in the Gaussian scale mixtures of Huser et al. (2017); other times, such as for the scale mixtures of log-Gaussian variables in Krupskii et al. (2016), or the model presented in Huser and Wadsworth (2018), this cannot be done. Where W is two-dimensional, the possible constructions stemming from (1) form an especially large class, since (W 1 , W 2 ) can itself have any copula. In this case, we particularly focus on how the multiplication by R changes the extremal dependence of (W 1 , W 2 ), summarized by the coefficients (χ W , η W ), to obtain the extremal dependence of the modified vector (X 1 , X 2 ) in terms of its coefficients (χ X , η X ). The marginal distributions of (W 1 , W 2 ) and R will play a crucial role, since, intuitively, the heavier the tail of R the more additional dependence is introduced in the vector (X 1 , X 2 ).
In this work we are focused on the upper tail of (X 1 , X 2 ), and as such we henceforth assume (W 1 , W 2 ) ∈ R 2 + ; observe that by the invariance of copulas to monotonic marginal transformations, we also therefore cover random location constructions of the form (Y 1 , Y 2 ) = S + (V 1 , V 2 ), S ∈ R, (V 1 , V 2 ) ∈ V ⊆ R 2 . For simplicity of presentation, we will often make the restriction that W 1 and W 2 have the same distribution, with comments on relaxations of this assumption given in Section 6. Furthermore, whilst our focus on the bivariate case permits simpler notation, the results are directly applicable to the bivariate margins of multivariate and spatial models, whose extremal dependence is typically analyzed in terms of the coefficients (2) and (3). Examples will be given in Section 4.
No widely recognized standard for ordering univariate tail decay rates in a stepless manner from the slowest to the fastest rate has emerged in the literature so far. A broad, but coarse, characterization is given by the three domains of attraction of the maximum. Specifically, we say that the random variable R is in the max-domain of attraction (MDA) of a generalized extreme value distribution if there exists a function b(t) > 0 such that as t → r = sup{r : P(R ≤ r) < 1}, P(R ≥ t + r/b(t))/P(R ≥ t) → (1 + ξr) for some ξ ∈ R, where a + = max(a, 0). The cases ξ > 0, ξ = 0, ξ < 0 define respectively the Fréchet, Gumbel and negative Weibull domains of attraction; the tail heaviness of R increases with ξ. However, the Gumbel limit in particular attracts distributions with highly diverse tail behavior such as finite upper bounds or heavy tails. Overall, this classification is therefore too coarse for our requirements, and it excludes important classes such as superheavy-tailed distributions defined through the property of heavy-tailed log-transformed random variables, which yield relevant constructions with random location. In addition to the maximum domains of attraction, we will utilize various commonly used tail classes, which are defined in Section 1.1; see also Appendix A. We begin in Section 2 by presenting results concerning the tail dependence of construction (1) according to the tail behavior of R and the shape of W, in the case where it is a one-dimensional support defined through a norm. We then characterize various cases where W is two-dimensional, according to the behavior of both R and (W 1 , W 2 ), in Section 3. Section 4 is devoted to literature review and framing a large number of existing examples in terms of our general results, whilst Section 5 illustrates the properties of some new examples inspired by the developments in the manuscript. In Section 6 we comment on generalizations and conclude. Proofs are presented in Section 7.

Terminology and notation
For a random variable Q, we define its survival function F Q (q) = P(Q ≥ q), and distribution function F Q (q) = 1 − F Q (q). If Q represents a bivariate random vector Q = (Q 1 , Q 2 ), we denote the minimum of its margins by Q ∧ = Q 1 ∧ Q 2 . For two functions f and g with g(x) = 0 for values x above some threshold value x 0 , we We recall definitions of upper tail behavior classes for a random variable X with distribution F . Abbreviations for distribution classes are chosen to be as self-explanatory as possible. Important tail parameters for these classes may be given as subscript, such as in ET α to refer to exponential-tailed distributions with rate α, but we may omit the subscript if the specific value of the parameter is not of interest. Relationships between distribution classes and membership of common distribution families are detailed in Appendix A.
Definition 2 (Regularly varying functions and distributions (RV 0 α and RV ∞ α )). A function g is regularly varying at infinity or at zero with index α ∈ R if g(tx)/g(t) → x α as t → ∞ or t → 0 respectively for any x > 0. We write g ∈ RV ∞ α or g ∈ RV 0 α respectively. If α = 0, then g is said to be slowly varying. A probability distribution Definition 3 (Exponential-tailed distributions (ET α and ET α,β )). The distribution F with upper endpoint as t → ∞, for any x > 0. If (5) holds with α = 0, we call F long-tailed, and we write F ∈ ET 0 . If α > 0 and By definition, F ∈ ET α with α ≥ 0 if and only if F (log(·)) ∈ RV ∞ −α . The class ET α,β with β > −1 is also referred to as gamma-tailed distributions. Another important subclass of ET α are the convolution-equivalent distributions.
We refer to the class CE 0 as subexponential distributions.

Constrained angular variables
In this section we focus on the case where W is defined by a norm ν; specifically we let W = {(w 1 , w 2 ) ∈ R 2 + : ν(w 1 , w 2 ) = 1}. Other types of constrained spaces may occasionally be of interest, however norm spheres are the most common restriction, and this focus allows greater generality in other aspects. In particular, all components of the vector are bounded in absolute value when the value of the norm is fixed. We then examine the extremal dependence based on the heaviness of the tail of R. Because the (W 1 , W 2 ) are bounded above, and subject to additional mild assumptions, we can and do classify R according to its MDA in this section.
The case where R belongs to the Fréchet MDA is the least delicate one, since as long as R has a much heavier tail than each of (W 1 , W 2 ), results do not depend strongly on other considerations. No equality in distribution is assumed between W 1 , W 2 in this case. When R is in the Gumbel or negative Weibull MDA, the shape of the norm ν becomes important, and some minor additional regularity conditions will be assumed, to be detailed in Section 2.2.

Radial variable in Fréchet MDA
Many of the most familiar results in the literature on extremal dependence concern the case when R is in the Fréchet MDA; this is equivalent to regular variation of the tail of R, namely F R ∈ RV ∞ −1/ξ , ξ > 0, where α = 1/ξ is called the tail index. A classical example of this is the multivariate Pareto copula, which can be constructed as in equation (1) with R standard Pareto, and W = {(w 1 , w 2 ) ∈ R 2 + : max(w 1 , w 2 ) = 1} (Ferreira and de Haan, 2014). Multivariate Pareto copulas can be identified with so-called extreme value copulas, which arise as the limiting copulas of suitably normalized componentwise maxima; see e.g. Rootzén et al. (2018). The next result provides the general form of the tail dependence coefficient for these models.
) < ∞, j = 1, 2, for some ε > 0. Then η X = 1, and Remark 1. When F R (r) ∼ Cr −α for some C > 0, then the condition E(W α+ε j ) < ∞ can be replaced by E(W α j ) < ∞, by Lemma 2.3 of Davis and Mikosch (2008). Remark 2. The condition E(W α+ε j ) < ∞ is guaranteed when W is the unit sphere of a norm ν; Proposition 1 notably also covers the case where (W 1 , W 2 ) ∈ R 2 + . Remark 3. The result includes the case α = 0, although the tail of such an R is too heavy to be in any domain of attraction. In this case, χ X = 1, representing perfect upper tail dependence. This case is discussed further in Section 3.1.
A more complete description of asymptotically dependent random vectors is given by the exponent function, defined as Small modifications to Proposition 1 yield The link between χ X and V X (1, 1) can be obtained simply by inclusion-exclusion arguments; in particular since min(a, b) = a + b − max(a, b), χ X = 2 − V X (1, 1).

Radial variable in Gumbel MDA
Suppose that R is in the Gumbel domain of attraction, with upper endpoint r i.e., where b(t) is termed the auxiliary function. One representation for distributions with this property is where z < r < r , c(r) → c > 0 as r → r , and a = 1/b is absolutely continuous with density a satisfying lim t→r a (t) = 0 (e.g Embrechts et al., 2013, Chapter 3.3). If r = ∞, we also have that lim r→∞ (rb(r)) ρ F R (λr)/F R (r) = 0 for any λ > 1, ρ ∈ R (Hashorva, 2012). Several distributions in this domain have mass on the negative half line, but we suppose here that R is conditioned to be positive, which does not affect the tail behavior. Notation and assumptions for (W 1 , W 2 ) and ν(W 1 , W 2 ) = 1. To this end, we assume that ν is a symmetric norm, i.e., ν(x, y) = ν(y, x), and scaled to satisfy ν(x, y) ≥ max(x, y), such that the unit sphere of ν is contained in that of max, with ν(b, 1 − b) = b for some b ≥ 1/2. Let τ (z) = z/ν(z, 1 − z) = 1/ν(1, 1/z − 1). For a random variable Z ∈ [0, 1], with distribution symmetric about 1/2, we can express Define as the interval such that τ (z) = 1 for all z ∈ I ν , and τ (z) < 1 for z ∈ I ν , and write with τ 1 strictly increasing and τ 2 strictly decreasing. Figure 1 illustrates τ for a particular ν; further illustrations are given in Appendix B. We assume further that (Z1): Z has a Lebesgue density, f Z , positive everywhere on [0, 1], and that its survival function is regularly varying at one, with F Z (1 − ·) ∈ RV 0 α Z , α Z > 0, and make the following mild regularity assumptions on the norm, ν, or equivalently τ .
(N1): The function τ is twice (piecewise) continuously differentiable except for finitely many points, at which we only require existence of left and right derivatives of first and second order.
We observe that asymptotic independence arises for ζ < 1, with the residual tail dependence coefficient determined by the properties of F R . The following corollary covers an important subclass of distributions in the Gumbel MDA.
Corollary 1. If ζ < 1 and − log F R ∈ RV ∞ δ , δ ≥ 0, then η X = ζ δ . We note that if δ = 0, such as in the case of log-normal R, η X = 1. Other possibilities in the Gumbel domain include − log F R (x) ∼ exp(x), as in the reverse Gumbel distribution, for which η X = 0. If r < ∞ then since the upper endpoint of X ∧ is less than that of X 1 , η X is not defined.
If ζ = 1, then one has asymptotic independence only if P(W = 1) = P(Z ∈ I ν ) > 0, which is equivalent to ν(x, y) behaving locally like the L ∞ norm around the point x = y. If ζ = 1 and P(W = 1) = 0, then we must have b 1 = b 2 = 1/2, and the "pointy" shape of such norms induces asymptotic dependence. The following example provides an illustration of this case.

Radial variable in negative Weibull MDA
Suppose that R > 0 is in the negative Weibull domain of attraction with upper endpoint r > 0, i.e, equivalently F R (r − ·) ∈ RV 0 α R . Note that the distribution of R cannot have a point mass on r . The general assumptions for (W 1 , W 2 ) are the same as in Section 2.2.

Unconstrained angular variables
We now treat the case where the support W is two-dimensional. As noted in Section 1, there are cases where (W 1 , W 2 ) itself might have a random scale representation, and by redefining the scaling variable we get back to the situation of one-dimensional W. We thus focus on constructions where this is not necessarily the case. To avoid additional complications we assume throughout this section that W 1 and W 2 share the common marginal distribution F W of the angular variable W . We also generally assume that the tail dependence coefficient χ W and the residual tail dependence coefficient η W of (W 1 , W 2 ) exist, although some results may still be obtained with the latter undefined. In Section 2, the constraints imposed by W being a unit sphere gave bounded marginal distributions for W j , j = 1, 2, and deterministic dependence between (W 1 , W 2 ). For two-dimensional W, the variety of marginal and dependence behaviors possible for (W 1 , W 2 ) means that systematic characterization according only to the MDA of R is more difficult. In fact, we need to consider different tail decays of both the radial variable R and the angular variable W since the combination of the two is crucial to classify the extremal dependence of (X 1 , X 2 ) = R(W 1 , W 2 ). We focus on some interesting sub-classes that still incorporate a wide variety of structures and cover most of the parametric univariate distributions available for R and W .
This section is structured according to the tail heaviness assumed for R, W , or both of them. In decreasing order we consider distributions with superheavy tails, regularly varying distributions, distributions of log-Weibull and Weibull type, and finally distributions with finite upper endpoint in the negative Weibull domain of attraction. Table 2 summarizes the general results developed in the following, and Table 3 contains the extremal dependence coefficients for all combinations of tail decays of R and W for the specific, yet interesting example where W 1 and W 2 are independent. Although we abstract away from marginal distributions in (X 1 , X 2 ), the interpretation of results from this section may sometimes be more natural in terms of S = log R and V = log W with random location structure, especially when S or V are heavy-tailed.

Superheavy-tailed variables
Suppose that R or W is superheavy-tailed, i.e., log R or log W is heavy-tailed, respectively. This case naturally arises when considering random location constructions log R + (log W 1 , log W 2 ), and we therefore additionally assume W > 0 so that log W j , j = 1, 2, are well defined. We will derive general tail dependence characterizations using classical results where convolution equivalence, and more specifically subexponentiality, is useful.
Radius R additional assumptions  Table 3: The values of χ X and η X for (X 1 , X 2 ) = R(W 1 , W 2 ) with W 1 , W 2 d = W independent, with different tail decay rates of the radial and angular variables. The *'s indicate that multiplication with R does not change the tail dependence of (W 1 , W 2 ), i.e., χ X = χ W = 0 and η X = η W = 1/2. The combinations of Weibull and log-Weibull tails remain open problems.

If
provided the limit exists.
Remark 4. The case c = 0 in point 1 of Proposition 4 is also covered by Remark 3 following Proposition 1, yielding χ X = 1. Proposition 4 extends this case to allow (W 1 , W 2 ) also to have superheavy tails.
Example 5 (Independence model). In order to illustrate the results of this section, we consider the example where R, W 1 and W 2 are independent. In this case and W has a comparable tail, then χ X ∈ (0, 1), whilst if W has tail lighter than superheavy, then χ X = 1. On the other hand, if W is superheavy-tailed with F log W ∈ CE 0 , then W ∧ is also superheavy-tailed. If R has lighter tail than W ∧ , then F R (x) ≤ CF W∧ (x) for large x with some C > 0, and by Proposition 4(2a) we have χ X = 0 and η X = 1/2. The case η X = η W may arise when the tail of W dominates the tail of R and the tail of R dominates the tail of W ∧ . For a concrete example, consider log-Weibull tails Then, η X = 1/(1 + c) according to Proposition 4(2b). The first row and column of Table 3 summarize these results.

Regularly varying variables
In this section we consider the case where R, W or both of them are regularly varying. When R is regularly varying with index α R > 0 and E(W α R +ε ) < ∞ for some ε > 0, then the tail dependence coefficient χ X is as described in Proposition 1 in Section 2.1. We firstly consider the case where W is regularly varying with index α W > 0 and R is lighter tailed, i.e., either also regularly varying with α R > α W or even lighter tailed such as distributions in the Gumbel or negative Weibull domain of attraction. Secondly, we study the case where both R and W are regularly varying with the same index α W = α R , which turns out to be particularly involved and which requires additional assumptions.
Proposition 5 (Regularly varying variables with W heavier than R). Let F W ∈ RV ∞ −α W , α W ≥ 0, and suppose that either F R ∈ RV ∞ −α R with α R > α W , or R is in the Gumbel or negative Weibull domain of attraction; denote the latter case by α R = +∞. Then χ X = χ W and The case where R and W are regularly varying with the same index α > 0 leads to various scenarios for the extremal dependence in (X 1 , X 2 ). Since F R , F W ∈ RV ∞ −α is equivalent to F log R , F log W ∈ ET α , and ET α is closed under convolutions, we have that F log X ∈ ET α (e.g., Lemma 2.5, Watanabe, 2008). Regarding the following results in Proposition 6, recall that the convolution equivalence property F log Y ∈ CE α implies finite moment E(Y α ) < ∞ for a random variable Y .
Proposition 6 (Regularly varying R and W with the same index). Let F R , F W ∈ RV ∞ −α with α > 0. Then: Remark 5. Proposition 6 contains certain results of Proposition 4 as a special case when allowing for α = 0. Proposition 6(2,3) treats the case of convolution-equivalent tails in log R or log W , which are relatively light since the expectation E(R α ) or E(W α ) respectively is finite; notice that ET α,β with β < −1 is an important subclass of CE α , see Lemma 2.3 of Pakes (2004). The tail of R is not dominated by that of W in Proposition 6(2), while it is dominated in Proposition 6(3). Proposition 6(4) shifts focus to relatively heavy tails in R with E(R α ) = ∞, such as the gamma tails of ET α,β with β > −1.
Example 6 (Independence model). We continue Example 5, where now W 1 and W 2 are independent and regularly varying with index α W , and R is regularly varying with index α R . We encounter different tail behavior of (X 1 , X 2 ) depending on the relation between α R and α W . If α R < α W , then we have asymptotic dependence with χ X given in (6). The same is true in general when W has a lighter tail than R that is not necessarily regularly varying. By Proposition 5, if α W < α R < 2α W , then (X 1 , X 2 ) is asymptotically independent with η X = α W /α R , and if α R > 2α W , then η X = 1/2. In general, if R is even lighter tailed, it does not affect the coefficients χ X and η X . If α R = α W = α, then η X = 1 and different scenarios for χ X can arise depending on the distributions of R and W , see Proposition 6. We consider α > 0. Since This fills the second row and column of Table 3.

Log-Weibull-type variables
In this and the following section we concentrate on radial and angular variables in the Gumbel domain of attraction. Due to the large variety of distributions in this domain we consider subsets that include the most commonly used distribution families. First, in this section, we study the case where both R and W are log-Weibull-tailed; equivalently, log R and log W are Weibull-tailed. We recall that a random variable Y is and we write F Y ∈ LWT β . The parameter β has the predominant influence on the tail decay rate, with β = 1 if and only if F Y ∈ RV ∞ −α , while β < 1 gives F Y ∈ SHT, and β > 1 yields rapid variation of Y , i.e., F Y ∈ RV ∞ −∞ ; these results can be shown by using a binomial series expansion for F Y (tx)/F Y (t). In the following, we denote the β-parameters of R and W by β R and β W , respectively. The superheavy-tailed case, β R < 1 or β W < 1, is already covered by Section 3.1, and the case of regularly varying tails with β R = 1 or β W = 1 is treated in Section 3.2.
In this section we study the remaining case β R > 1 and β W > 1, which encompasses important radial and angular distributions such as the log-Gaussian ones. Similarly to Section 3.1, it makes sense to log-transform the random scale construction R(W 1 , W 2 ) to obtain the random location construction log R + log(W 1 , W 2 ) where we can apply convolution-based results. When independent heavy-tailed summands are involved in the convolution typically only one of the values of summands has a dominant contribution to a high values of the sum, resulting in relatively simple formulas; see Section 3.1. On the contrary, in the light-tailed setup all summands may contribute significantly when high values arise in the sum, rendering the tail analysis more intricate. Only relatively few general results on convolutions with tails lighter than exponential are available in the literature.
The following lemma will be useful for this and the next section.
provided the limit exists.
3. If β W∧ > β W , then log F W = o(log F W∧ ), and η W is not defined.
Remark 6. The proof of the above lemma is straightforward from (17). It also covers the case where (W 1 , W 2 ) is a bivariate random vector with W and W ∧ are log-Weibull-tailed, since the tail dependence coefficient and the residual tail dependence coefficient are invariant under monotonic marginal transformations.
We consider the set-up where the components R, W and W ∧ are log-Weibull-tailed with the same coefficient β > 1 and a simplified form of the slowly varying function by assuming that it is asymptotically constant, i.e., (x) ∼ c > 0.
Proposition 7 (Light-tailed random location with F R ∈ LWT β , β > 1). Suppose that F R , F W , F W∧ ∈ LWT β with possibly different parameters α, γ indexed by the corresponding R, W and W ∧ , but where β = β R = β W = β W∧ > 1. Assume that the slowly varying functions behave asymptotically like positive constants. If χ W > 0, then If χ W = 0, then χ X = 0 and where Example 7 (Gaussian factor model). Suppose that log R is univariate standard Gaussian and that log(W 1 , W 2 ) is bivariate standard Gaussian, independent of R and with Gaussian correlation ρ W ∈ (−1, 1]. Then we have log-Weibull tails with parameters β R = β W = β W∧ = 2, α R = α W = 1/2 and α W∧ = 1/(1 + ρ W ) (see Example 9). Applying (20) gives Example 8 (Independence model). As in Examples 5 and 6 we let R, W 1 and W 2 be independent, and we now assume that they are log-Weibull-tailed with equal β parameter. Using the independence assumption we observe that F W∧ ∈ LWT β with α W∧ = 2α W . Proposition 7 implies that (X 1 , X 2 ) is asymptotically independent, and the residual tail dependence coefficient η X can be calculated by formula (20).

Weibull-type variables
In this section we consider the case where R and W follow a Weibull-type distribution, a rich class in the Well-known examples of Weibull-tailed distributions are the Gaussian with β = 2, the gamma with β = 1 or, more generally, the Weibull where β is called the Weibull index. For developing useful results, we further assume that, in addition to R and W , W ∧ also has a Weibull-type tail. As previously, we index the corresponding functions and the parameters α, γ in (21) by the variable name. We also recall Lemma 1 concerning the dependence coefficients of the vector (W 1 , W 2 ).
We have the following hierarchy of dependence structures: if the limit exists, and η X = η W = 1.
In all of the cases encompassed by Proposition 8, (X 1 , X 2 ) and (W 1 , W 2 ) have the same tail dependence coefficient χ, which can be positive only in case 1. In all other cases the variables are asymptotically independent, and only in case 3 the residual tail dependence coefficient η changes under the multiplication of the radial variable R. Since β R /(β R +β W ) ∈ (0, 1), this multiplication always leads to an increase in dependence, that is, η X > η W .
Example 9 (Gaussian scale mixtures). To illustrate the most interesting case 3 in the above proposition we consider (W 1 , W 2 ) following a bivariate normal distribution with standardized margins and correlation ρ W . We have that where the tail distribution of the minimum follows from bounds on the multivariate Mills ratio, see e.g. Hashorva and Hüsler (2003), and r W and r W∧ regularly varying functions of order −1 and −2 respectively. Therefore, η W = (1 + ρ W )/2 and Proposition 8 yields , confirming Theorem 2 in Huser et al. (2017).
Example 10 (Independence model). We continue the example where R, W 1 and W 2 are independent, and they are now assumed to be Weibull-tailed, where the latter two have the same distribution as W . By independence, W ∧ is also Weibull-tailed with β W∧ = β W and α W∧ = 2α W . Therefore, the third part of Proposition 8 entails that η X = 2 −β R /(β R +β W ) . Note that this expression tends to 1/2 if β R /β W tends to infinity such that the tail of W dominates strongly with respect to that of R; if β R /β W tends to 0, then η X tends to 1.

Variables in the negative Weibull domain of attraction
The remaining cases are those where R, W or both of them have finite upper endpoint and are in the negative Weibull domain of attraction. We recall that a variable Y is in the negative Weibull domain of attraction if The case where R is superheavy-tailed or regularly varying and W satisfies (24) has been covered in part 1 of Proposition 4 and Proposition 1, respectively. On the other hand, the case where the tail of R satisfies (24) and W is superheavy-tailed or regularly varying is treated by part 2 of Proposition 4 and Proposition 5, respectively. It remains to study the situation where one of R or W is of form (24), and the other is in the Gumbel domain of attraction as defined in (9). In this section we focus on the case where W ∧ has the same upper endpoint as W ; for a more detailed study where W ∧ can have a smaller upper endpoint, and W may have a point mass on its upper endpoint, see Section 2.
Proposition 9 (Variables in the negative Weibull domain of attraction).
1. Suppose that R is in the Gumbel MDA with upper endpoint r and that W and W ∧ satisfy (24) with parameters α W and α W∧ , respectively. Then χ X = χ W and η X = 1. (24) and let W and W ∧ be in the Gumbel MDA with equal upper endpoint w ∈ (0, ∞] and auxiliary functions b W and b W∧ , such that lim x→w b W (x)/b W∧ (x) exists. Then χ X = χ W and η X = η W .

Let R satisfy
3. Let R, W and W ∧ all satisfy (24) with endpoints r , w , w and parameters α R , α W and α W∧ , respectively. If α W∧ = α W then χ X = χ W and η X = 1. If α W∧ > α W then χ X = χ W = 0 and Example 11 (Independence model). Let us consider again the simple model where R, W 1 d = W 2 are independent, with F R , F W satisfying (24) with parameters α R , α W . Clearly, F W∧ also satisfies (24), with parameter α W∧ = 2α W . The third part of Proposition 9 shows that hence by varying the parameters α R , α W > 0 we can attain the whole range of residual tail dependence coefficients related to positive association.

Literature review and examples
Here we present an overview of related literature, detailing how existing examples and results fit into the framework of this paper.
Archimedean copulas can be viewed as a special case of Liouville copulas, whose dependence properties are studied in Belzile and Nešlehová (2018). For (X 1 , X 2 ), their Theorem 1 states that R in the Fréchet MDA leads to asymptotic dependence, whilst the Gumbel and negative Weibull maximum domains of attraction lead to asymptotic independence. The exponent function given in their Theorem 1 matches equation (7). In their Theorem 2, Belzile and Nešlehová (2018) consider the extremal dependence properties of 1/(X 1 , X 2 ) =R(W 1 ,W 2 ), i.e., the Liouville copula itself. Since the reciprocal of Dirichlet random variables have regularly varying tails, this links with Proposition 5 which states that asymptotic independence arises if (W 1 ,W 2 ) themselves are asymptotically independent and heavier-tailed than R. Proposition 6(4c) is relevant ifR andW are regularly varying with the same index.
Example (Bivariate Pareto copula associated to the Gumbel copula). Since the Gumbel copula is a max-stable distribution, it has an associated Pareto copula. If Z has density f Z (z) = h(z) max(z, 1 − z)2 1−θ , where h is given by (8), then (X 1 , X 2 ) = R(W 1 , W 2 ) = R(Z, 1−Z)/ max(Z, 1−Z) with F R (r) = r −1 leads to the associated bivariate Pareto copula. The distribution function of (X 1 , X 2 ) is ) θ is the exponent function for the logistic / Gumbel distribution.
Model of Wadsworth et al. (2017) They consider the copula induced by taking R to be generalized Pareto, F R (r) = (1 + λr) −1/λ + , and W = {(w 1 , w 2 ) ∈ [0, 1] 2 : (w 1 , w 2 ) * = 1} where · * is a symmetric norm subject to certain restrictions. These restrictions mean that λ ≤ 0 corresponds to asymptotic independence; the residual tail dependence coefficient η X is as given in Proposition 2 for λ = 0 with ζ = τ (1/2) = (1, 1) −1 * , and Proposition 3 for λ < 0. Here we note that if the norm ν has certain shapes that were excluded in Wadsworth et al. (2017), asymptotic dependence is possible for λ ≤ 0. When R is in the Fréchet MDA (λ > 0) then asymptotic dependence holds with χ X given by (6). Krupskii et al. (2016) They consider location mixtures of Gaussian distributions, corresponding to scale mixtures of log-Gaussian distributions. According to their Proposition 1, asymptotic dependence occurs when the location variable is of exponential type, i.e. the scale is of Pareto type; the given χ X can then be obtained via (6). When the location is Weibull-tailed but with shape in (0, 1), the scale is superheavy-tailed, with F R ∈ RV ∞ 0 , and perfect extremal dependence (χ X = 1) arises, as noted in Remark 3 following Proposition 1. When the random location is Weibull-tailed with shape in (1, ∞) then the random scale R is in the Gumbel MDA and asymptotic independence arises. If F log R ∈ WT 2 has the same Weibull coefficient 2 as the standard Gaussian log W and as the minimum of log W 1 and log W 2 (provided that their linear correlation is in (−1, 1]), then we can apply Proposition 7 to calculate the value of η X given as

Model of
which extends the results of Krupskii et al. (2016). Specifically, with standard Gaussian log R we get η X = (3 + ρ)/4, see Example 7. (2018) They consider scale mixtures of asymptotically independent vectors where both R and W have Pareto margins with different shape parameters. Asymptotic dependence arises when R is heavier tailed; χ X is then given by (6), whilst asymptotic independence arises when W is heavier tailed and η X is given by (15). When R and W have the same shape parameter, the assumption η W < 1 implies that E(W α+ε ∧ ) < ∞ for some ε > 0, hence Proposition 6 (4c) gives asymptotic independence.

Model of Huser and Wadsworth
Various other articles also focus on polar or scale-mixture representations. Hashorva (2012) examines the extremal behavior of scale mixtures when R is in the Gumbel MDA. He considers both one-and two-dimensional W, both with similarities and differences to the set-up herein. For one-dimensional W, he supposes a functional constraint of the form (W 1 , W 2 ) = (W, ρW + z * (W )) for measurable z * : [0, 1] → (0, ∞); ρ ∈ (−1, 1). (In fact, the specification also allows for negative components, but we focus here on the positive part.) Constraints to certain norm spheres, such as the Mahalanobis or L p norm, could be written in this way, however examples such as the L ∞ norm do not admit this representation. Where the representations overlap, our results coincide (e.g., Section 4.3 of Hashorva (2012)). In the case of two-dimensional W, W 1 is assumed bounded, whilst W ∧ is in the negative Weibull MDA. Although the framework of Hashorva (2012) does not assume the symmetry of our framework, there are nonetheless some connections between the results in our Section 3.5 and that paper. Nolde (2014) provides an interpretation of extremal dependence in terms of a gauge function (see also Balkema and Nolde (2010)), which, loosely speaking, corresponds to level sets of the density in light-tailed margins. Their main result (Theorem 2.1) is presented in terms of Weibull-type margins, such that − log F X ∈ RV ∞ δ , δ > 0; in terms of Section 2, this corresponds to − log F R ∈ RV ∞ δ . By noting that where the density of R, f R , exists, the joint density of (X 1 , X 2 ) is the gauge function of (X 1 , X 2 ) is obtained as ν when − log F R ∈ RV ∞ δ , using Proposition 3.1 therein. We found that η X = ζ δ , with ζ = ν(1, 1) −1 , precisely as given in Nolde (2014).
Various papers focus on extremal dependence arising from certain types of polar representation, but from a conditional extremes perspective (Heffernan and Tawn, 2004;Heffernan and Resnick, 2007). This is different to our focus; here we examine the extremal dependence as both variables grow at the same rate. In the conditional approach, different rates of growth may be required in the different components. Articles taking this perspective include Abdous et al. (2005) in the context of elliptical copulas, whilst Fougères and Soulier (2010) and Seifert (2014) consider the constrained W case, with R in the Gumbel MDA.

New examples
We present two new constructions that have the desirable property of smoothly interpolating between asymptotic dependence and asymptotic independence, whilst yielding non-trivial structures within each class. By smoothness, we mean that the transition between classes occurs at an interior point, θ 0 , of the parameter space Θ, and, assuming increasing dependence with θ, lim θ→θ0+ χ X = 0, lim θ→θ0− η X = 1. To our knowledge, the only other models in the literature with this behavior are (i) that of Wadsworth et al. (2017), where ν(x, y) = max(x, y) and F R (x) = (1 + λx) −1/λ + , λ ∈ R, and (ii) that of Huser and Wadsworth (2018) where , δ ∈ (0, 1), and η W < 1. The first example is constructed using constrained (W 1 , W 2 ) (Section 2), where the required ingredients are F R , ν, and F Z , whilst the second uses unconstrained (W 1 , W 2 ) (Section 3) with ingredients F R , F W and the dependence structure of (W 1 , W 2 ).

Model 1
In Propositions 2 and 3, it was demonstrated how the shape of ν affects the tail dependence of (X 1 , X 2 ) when R is in the Gumbel or negative Weibull MDA. In Example 2, a particular norm that yields asymptotic dependence was given; here we extend the parameterization of this norm and use our results to present a new asymptotically (in)dependent copula. Since the cases where R has finite upper endpoint r < ∞ lead to undefined η X , we focus on r = ∞.
To make things concrete, we propose the following model.
The set of models defined in Proposition 10, exemplified by Model 1, has some rather interesting behavior in the extremes. Whilst the limiting quantities χ X , η X are given in Proposition 10, the subasymptotic behavior of (X 1 , X 2 ), in particular the behavior of the slowly varying function in (2), is not prescribed by any of the propositions in this paper. Combining equations (2) and (3), define so that for η X = 1, χ X (q) = (1 − q). For Model 1 we find that χ X (q) is not necessarily monotonic, and may decrease before increasing to a positive limit value. Figure 5.1 shows χ X (q) plotted against q for various parameterizations of the model. This non-monotonic behavior is seemingly uncommon; to our knowledge there are no well-known theoretical examples of this.

Model 2
The following proposition collates results from Propositions 1 and 9, and provides a general principle for constructing new dependence models permitting both asymptotic dependence and asymptotic independence.
Proposition 11. Let R be in the MDA of a generalized extreme value distribution with shape parameter ξ ∈ R, and let (W 1 , W 2 ) with W 1 d = W 2 d = W have χ W = 0, well-defined η W ∈ (0, 1), and F W (w − ·) ∈ RV 0 α W , α W > 0. Then The model construction opportunities from Proposition 11 are quite varied; specifically taking F R that permits all three tail behaviors produces a flexible range of models spanning the two dependence classes. We therefore propose the following concrete model, based on our running independence example.

Discussion
The paper studies the extremal properties of copulas (X 1 , X 2 ) = R(W 1 , W 2 ) and determines the coefficients χ X for asymptotic dependence and η X for asymptotic independence, respectively.
In the case of constrained (W 1 , W 2 ) defined on the sphere corresponding to some norm ν, in Section 2, the extremal dependence is characterized in terms of the tail heaviness of R. Classical results on multivariate Pareto copulas are recovered for regularly varying R, whereas new structures are obtained for distributions of R with light tails or finite upper endpoint. For the Gumbel domain of attraction, in particular, we get a large variety of behaviors for asymptotically independent (X 1 , X 2 ) that strongly depend on the auxiliary function b of R and the shape of the ν-sphere. This extends the results of Wadsworth et al. (2017) who considered only the exponential distribution in this class.
For unconstrained distributions of both R and W , Section 3 formalizes the general intuition that heavier tails of R introduce more additional dependence in (X 1 , X 2 ). The results summarized in Table 3 for the special case of the independence model allow for several conclusions. The most interesting (and involved) situations figure along the main diagonal where R and W have similar tail behavior. Above this main diagonal, R is so heavy that it mostly dominates the extremal dependence in (X 1 , X 2 ). On the other hand, below the diagonal, R is too light tailed, relatively to W , to have an impact on the tail dependence coefficients χ X and η X . Similar observations hold true for the more general case of arbitrary dependence in (W 1 , W 2 ) summarized in Table 2.
We note that there is a clear overlap between the results obtained in Sections 2 and 3. If one considers χ W as derived from the shape of W, then many results in Section 2 are obtained from Section 3, just as Proposition 1 is relevant in both sections. However, the separate treatment seems justified on the grounds of the importance of such constructions, and the additional insight gained in focusing on the shape of W.
The above results provide a general and unifying framework to analyze bivariate extremal dependence, and Section 4 shows that they cover many of the existing examples in the copula and the extreme value literature. Most importantly, combining the insights from different sections enables the construction of numerous new statistical models that smoothly interpolate between asymptotic dependence and independence; see Section 5 for two instances. Although our focus was on dependence, knowledge on how the marginal scales of R and W and the dependence properties of (W 1 , W 2 ) influence the dependence of (X 1 , X 2 ) makes it easier to construct models (X 1 , X 2 ) that naturally accommodate both marginal distributions and dependence of multivariate data. Such modeling avoids what may be construed as the artificial separation of modeling of margins and dependence known as copula modeling, which has been strongly criticized by some authors since it may lack interpretation with respect to the data generating mechanism (Mikosch, 2006). For example, in factor constructions based on independent random variables, such as the ones with independent W 1 and W 2 discussed throughout, our results give guidance on the relative tail heaviness of R with respect to (W 1 , W 2 ) necessary to transition from asymptotic independence to asymptotic dependence in (X 1 , X 2 ), and both heavy-or light-tailed marginal distributions are possible by considering the distribution of either (X 1 , X 2 ) or log(X 1 , X 2 ) as a model for data.
In Sections 2 and 3, we often considered the simplification W 1 d = W 2 , yielding X 1 d = X 2 , which allows the coefficients χ X and η X to be calculated without reference to marginal quantile functions. A weaker sufficient condition for this is F X1 (x) ∼ F X2 (x) as x → x , with x a common upper end point. To see this sufficiency, define x q = min{F −1 X1 (q), F −1 X2 (q)}, x q = max{F −1 X1 (q), F −1 X2 (q)}, and note that where max{F X1 (x q ), F X2 (x q )} = min{F X1 (x q ), F X2 (x q )} = 1−q. Consequently, the tail dependence coefficient χ X of (X 1 , X 2 ), if it exists, is bounded between the limit superior of the left-hand side and the limit inferior of the right-hand side in (28), respectively, for q → 1. Whilst these bounds hold in general for common upper end point, they deliver the precise coefficient χ X only if F X1 (x) ∼ F X2 (x), x → x or both limits are zero. Similar arguments apply to the residual tail dependence coefficient η X , where the corresponding bounds determine η X under the weaker requirement log F X1 (x) ∼ log F X2 (x), x → x . Whilst our focus has been on the coefficients χ X and η X , we note that there are important aspects of the dependence structure that are not described by these coefficients. For example, in Section 2, we found that when R was in the Gumbel or negative Weibull MDA, χ X and η X depended only on the shape of ν and the distribution of R, but not at all on the distribution of Z. Nonetheless, the latter plays an important role in the behavior of the slowly varying function in (3), which was exemplified in Figure 5.1.

Proofs
We recall Breiman's lemma, which will be useful in several contexts.
We also note that in the case of common marginal distributions F X with upper endpoint x , the tail dependence coefficient in (2) is χ X = lim x→x F X∧ (x)/F X (x), whilst the residual tail dependence coefficient in (3) is given by η X = lim x→x log F X (x)/ log F X∧ (x).

Proofs for Section 2
Proof of Proposition 1. Since E(W α+ε j ) < ∞, j = 1, 2, Lemma 2 gives so that F Xj ∈ RV ∞ −α . Now consider the quantile functions of X j and R; denote these by F −1 Xj (q), F −1 R (q). Suppose firstly that α > 0. By taking the reciprocal of relation (29), and using Proposition 2.6 (vi) of Resnick (2007), we have Consider now Since E(W α+ε j ) < ∞, j = 1, 2, and min( and so as q → 1 from which the result follows. For α = 0 we have F Xj (x) ∼ F R (x) ∈ RV ∞ 0 , as well as F X∧ (x) ∼ F R (x). Using the bounds in (28) and taking limits, we get χ X = 1. As noted after the proposition, the conditions ensure χ X > 0 and hence η X = 1.
Before proceeding to the proofs of Propositions 2 and 3, Lemma 3 provides further detail about the function τ , whilst Lemma 4 details the behavior of W and W ∧ .
For the proofs of Propositions 2 and 3 we will repeatedly use Theorem 3.1 of Hashorva et al. (2010), and so restate the relevant components here as a lemma.
Proof of Proposition 3. Following a similar line to Proposition 2, the marginal distribution satisfies for a new slowly varying function˜ , whilst for the joint distribution If ζ < 1, then X ∧ has upper bound ζr < r and so χ X = 0, with η X not defined. If ζ = 1 and P(W = 1) > 0, then χ X = 0, and Otherwise, if ζ = 1 and P(W = 1) = 0, then since α W = 1 with the behavior of and ∧ as given in (31) and (32).

Proofs for Section 3
For simplicity of notation, we often write S = log R and V = log W and V ∧ = log W ∧ in the following. We denote the common marginal distribution of X 1 and X 2 by F X . We start by recalling an important result from the literature on the convolution with a convolution-equivalent distribution.
Lemma 6 (Convolution with a distribution in CE α , see Theorem 1 of Cline (1986) and Lemma 5.1 of Pakes (2004)). Let Y 1 , Y 2 be two random variables with distributions F 1 , F 2 respectively. If F 1 ∈ CE α with α ≥ 0 and The following lemma establishes the link between convolutions of nonnegative exponential-tailed distributions and the case where negative values may arise; it will be useful in several contexts.
Lemma 7 (Convolutions of exponential-tailed distributions with negative values). For i = 1, 2 and probability distributions F i ∈ ET α defined over R with α > 0, denote p i = F i (0) ∈ [0, 1] the probability of nonnegative values. Using the convention 0/0 = 0, let x < 0, denote the conditional distribution of F i over negative values. We use the notation M H (α) = ∞ −∞ exp(αy)H(dy) for a given distribution H. Then: Specifically, if c 1 = M F + 2 (α) and c 2 = M F + 1 (α), then and if c 1 = M F + 2 (α) and c 2 = 0, then Proof. We start with the mixture representation We can then use the equation p i F + i (x) = F i (x), x ≥ 0, and the following inequalities for {i 1 , i 2 } = {1, 2}, Moreover, Lemma 2 can be applied for mixed terms, yielding , and then (42) and (43) follow from straightforward calculations. To determine the behavior for special cases of c 1 and c 2 , observe that We then apply (41) and obtain , and the value χ X in (13) follows. Since χ X > 0 in all cases, we have η X = 1. (41) gives F X ∼ F W . If χ W > 0, then F S = o(F V∧ ), and (41) yields the tail equivalence F X∧ (x) ∼ F W∧ (x), which entails χ X = χ W . In the case where χ W = 0, we have F V∧ = o(F V ) and F S = o(F V ), such that Foss et al. (2009, Corollary5(ii)) establishes F X∧ ∼ F R + F W∧ + o(F W ), and χ X = χ W = 0 follows. 2.a) When further F S , F V∧ ∈ CE 0 and F R (x)/F W∧ (x) ≤ C with C > 0 as x → ∞, we consider the two boundary cases F R = o(F W∧ ) and F R /F W∧ ∼ C. Using (41), we get

Let
in both cases, which proves η X = η W . 2.b) When further F S ∈ CE 0 and F W∧ = o(F R ), then we get F X∧ (x) ∼ F R (x) from (41), such that x → ∞, and the limit is η X if it exists.
Proof of Proposition 5. Since E(R α W +ε ) < ∞ for a small ε > 0, by Lemma 2, the marginal distributions satisfy so that X is also regularly varying with index α W . The coefficient η W is not defined if W ∧ has a finite upper endpoint, in which case E(W α R +ε ∧ ) < ∞. Otherwise, we have that W ∧ is regularly varying with index α W /η W , or if η W = 0, F W∧ decays faster than any power. Again, by Lemma 2, we obtain that If χ W > 0 then η W = 1, so we conclude that For the coefficient of residual tail dependence between X 1 and X 2 we obtain Before proceeding to the proof of Proposition 6, we recall several lemmas from the literature.
Lemma 8 (Convolution of distributions in ET α,β>−1 , see Theorem 4(v) of Cline (1986)). For two distributions F i ∈ ET α,βi , i = 1, 2, possessing gamma-type tail with slowly varying i (x), we get Proof. The result for nonnegative S and V is found in Theorem 4(v) of Cline (1986). The extension to negative values then follows from Lemma 7(1).
Lemma 9 (Ratio of convolutions with a distribution in ET α ). Let F ∈ ET α with α > 0, and let G 1 , G 2 be distributions satisfying Proof. The result is given in Theorem 6(ii,iii) of Cline (1986) for F (0) = 1 and G 1 (0) = 1. For point 1, the extension to negative values in F and G 1 follows from observing that Theorem 6(iii) of Cline (1986) implies where F + is obtained from F by setting F + (0) = F (0) and F + (0) = 1, and the same construction is taken for G + 1 ; i.e., F + and G + 1 arise from projecting negative values to 0. For point 2, the extension to negative values can be shown using Lemma 7(2). Indeed, the same limit M G1 (α) arises for F G 1 (x)/F (x) if we project negative values in F and G 1 to 0 or not.
It remains to show that the limit in (50) does not change when we substitute log F S+V∧ (x) for log F S (x) in the denominator, where no additional assumption on the distribution of V ∧ is made.
2. In the case where F S ∈ CE α and F V (x)/F S (x) → c ≥ 0, x → ∞, we can use Lemma 6 with F 1 = F S and F 2 = F V , which yields and by setting F 2 = F V∧ , we get Combining these two results yields the value of χ X .
3. If F V ∈ CE α and F S (x)/F V (x) → 0, x → ∞, we use Lemma 6 to show F S+V (x)/F V (x) ∼ E(e αS ) = E(R α ). If χ W > 0, then also F V∧ ∈ CE α and we have F S+V∧ (x)/F V∧ (x) ∼ E(R α ) by analogy. By combining these two results, we get Consider the copula (V 1 , V 2 ) defined as the mixture of (V 1 , V 2 ) and (V, V ) with probabilities (1 − ) and , respectively, for some 0 < < 1. The marginal distribution of (V 1 , V 2 ) is still F V , and the induced V ∧ satisfies Since V ∧ is stochastically larger than V ∧ , this means that also Y + V ∧ is stochastically larger than Y + V ∧ for any random variable Y , by a coupling argument. Thus, where the last but one equation follows from the former case where χ W > 0 (with V ∧ taking the role of V ∧ ).
Since > 0 is arbitrary, the result follows.
Using Lemma 10 for X = RW , we find that F X ∈ WT β X with parameters β X = β R β W /(β R + β W ), and slowly varying function X > 0. The same result applies to X ∧ = RW ∧ with constants β X∧ , α X∧ , γ X∧ , and slowly varying X∧ > 0.
1. In this case, it follows from Lemma 10 that also β X∧ = β X , α X∧ = α X , and γ X∧ = γ X , and thus since all other dominating terms of higher order cancel out, and the further results follow straightforwardly.
Proof of Proposition 9. 1. Applying Lemma 5 to compute F X (x) and F X∧ (x) gives since xb R (x) → ∞ for x → r , α W∧ ≥ α W , and if χ W > 0 then necessarily α W∧ = α W . Similarly, η X = lim x→r log F X (x)/ log F X∧ (x) = 1, since, by l'Hôpital's rule lim x→r log(xb R (x))/ log F R (x) = 0, and therefore the term log F R (x) dominates both the numerator and the denominator.
Proof of Proposition 11. The first claim follows directly from Proposition 1, since E(W 1/ξ+ε ) < ∞. Since η W ∈ (0, 1) is defined, W ∧ has the same upper endpoint as W , and F W∧ (w − s) = W∧ (F W (s))F W (s) 1/η W , implying W ∧ is also in the negative Weibull MDA with α W∧ = α W /η W . The second and third claims then follow from parts 1 and 3 of Proposition 9.

A Tail classes and examples
Definitions of tail classes are given in Section 1.1. The following lemma summarizes important relationships between such tail classes.
We recall the membership in tail classes for well-known parametric distribution families in Table 4, see Johnson et al. (1994Johnson et al. ( , 1995 for reference about parameters. Here we abstract away from the usual parameter symbols of these distributions to avoid conflicting notations with general tail parameters. We refer parameters as scl and loc if scl × X + loc has scale scl and location loc, where X has scale 1 and location 0. Another parameter shp may be related to shape for some distributions.