New characterizations of multivariate Max-domain of attraction and D-Norms

In this paper we derive new results on multivariate extremes and D-norms. In particular we establish new characterizations of the multivariate max-domain of attraction property. The limit distribution of certain multivariate exceedances above high thresholds is derived, and the distribution of that generator of a D-norm on ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb R}^{d}$\end{document}, whose components sum up to d, is obtained. Finally we introduce exchangeable D-norms and show that the set of exchangeable D-norms is a simplex.


Introduction
Multivariate extreme value theory (MEVT) is the appropriate toolbox for analyzing several extremal events simultaneously. However, MEVT is by no means easy to access; its key results are formulated in a measure theoretic setup in which a common thread is not visible.
Writing the 'angular measure' in MEVT in terms of a random vector, however, provides the missing common thread: Every result in MEVT, every relevant probability distribution, be it a max-stable one or a generalized Pareto distribution, every Michael Falk michael.falk@uni-wuerzburg.de Timo Fuller timo.fuller@uni-wuerzburg.de 1 relevant copula, every tail dependence coefficient etc. can be formulated using a particular kind of norm on multivariate Euclidean space, called a D-norm. Deep results like Takahashi's characterizations of multivariate max-stable distributions with independent or completely dependent margins (Takahashi 1987;1988) turn out to be elementary and easily seen properties of D-norms. The letter D means dependence, because a D-norm describes the dependence among the margins of a multivariate max-stable distribution, see Theorem 4.1.
Norms are introduced in each introductory course on mathematics as soon as the multivariate Euclidean space is introduced. The definition of an arbitrary D-norm requires only the additional knowledge of random variables and their expectations. But D-norms do not only constitute the common thread through MEVT; they are of mathematical interest of their own. D-norms were first mentioned in Falk et al. (2004), equation (4.25), and more elaborated in Falk et al. (2011), Section 4.4. But it was recognized only later that D-norms are actually the skeleton of MEVT and that they simultaneously provide a mathematical topic, which can be studied independently. The monograph Falk (2019) compiles the contemporary knowledge about D-norms and provides an introductory tour through the essentials of MEVT. But D-norms can also be seen from a functional analysis perspective as in Ressel (2013), which is Section 1.11 in Falk (2019), or from a stochastic geometry point of view as in Molchanov (2008), presented in detail in Section 1.12 in Falk (2019).
In this paper we establish new results on MEVT and D-norms. In Section 2 we recall the definition of D-norms and list several basic facts. In Section 3 we specify functions 0 0 , such that the number is generator invariant, i.e., depends only on the underlying D-norm generated by , but not on the particular generator . The dual D-norm function is a prominent example. But this result also entails the definition of the co-extremality coefficient, which is a measure of pairwise tail dependence of a multivariate distribution. The corresponding matrix of co-extremality coefficients turns out to be positive semidefinite.
In Section 4 we link D-norms with MEVT and establish particularly in Theorem 4.4 a new characterization of max-domain of attraction of a random vector 1 . An immediate consequence, in Corollary 4.6, is for example the fact that the probability of the event 1 is approximately as t increases, independent of the dependence structure among 1 ; this was already observed in Barbe et al. (2006).
In Section 5 we derive the limit distribution of certain multivariate exceedances above high thresholds. These results are used in Section 6 to derive among others the distribution of that generator of a D-norm on , whose components sum up to d.
It turns out that Mai and Scherer (2020), which is on exchangeable extreme-value copulas, is actually a contribution to the theory of D-norms, as shown in Section 7. To the best of our knowledge, the concept of an exchangeable D-norm is introduced here for the first time, and we prove new results for this subfamily of D-norms.

D-Norms
The following result introduces D-norms. For a proof we refer to Lemma 1. There is a smallest D-norm and a largest one: 1 see equation (1.4) in Falk (2019). Note that the univariate distributions of the components 1 of influence the D-norm, which is generated by . Suppose 1 are independent and identically distributed (iid) random variables. Then it makes a difference, whether they follow the standard exponential distribution or the uniform distribution on (0,2).
Neither the generator of a D-norm is uniquely determined nor its distribution. Actually, if 1 is the generator of a D-norm, then 1 2 generates the same D-norm if X is a random variable with 0 and 1, which is also independent of . But for any D-norm on and an arbitrary norm on there exists a generator of with the additional property const. The distribution of this generator is uniquely determined. This is the content of the following result, which is Theorem 1.7.1 in Falk (2019).

Theorem 2.3 (Normed Generators) Let
be an arbitrary norm on . For any D-norm on , there exists a generator with the additional property const. The distribution of this generator is uniquely determined.
The following characterization will be used in several proofs in this paper. It goes back to Takahashi (1987) and Takahashi (1988). For a proof see Corollary 1.3.5 in Falk (2019).

Corollary 2.5 Let
be an arbitrary D-norm on . We have the characterizations:

When the number E(h(Z)) is generator invariant
Let be the generator of a D-norm on , and let 0 0 be a continuous function that is homogeneous of order one, i.e., , 0, 0 . In Theorem 3.5 we will establish the fact that, with such a function h, the number does not depend on the particular generator of . Take, for example, an arbitrary norm on . With we obtain that the number does not depend on the particular generator of a D-norm . Equally, with min 1 , Theorem 3.5 explains, why the function min 1 , known as the dual D-norm function, is generator invariant.
Our proof of Theorem 3.5 uses several auxiliary results, which we establish first. They might be of interest of their own. We will frequently use the equation 0 (3.1) valid for an arbitrary random variable 0; for a proof see, e.g., Lemma 1.2 in Falk (2019). By 1 we denote the indicator function of a set A, i.e., 1 1 if , and zero elsewhere. All operations on vectors are meant componentwise. The measure , introduced in the next result, is known as the exponent measure in MEVT; see, for example, Section 4 in Falk (2008). The definition of the exponent measure in equation (1.17) in Falk (2019) is, however, restricted to generators that realize in some unit sphere. Different to that, the measure in Eq. 3.3 supposes no restriction on . This general definition is particulary required for the derivation of Theorem 3.5 below.

Proposition 3.2 Choose a D-norm
on and an arbitrary generator of . Put 0 0 and define the function 0 by 1 0 .
defines a measure on the Borel -field of with where denotes the Lebesgue measure on 0 , is the distribution of , and " " denotes the product measure. The measure on is uniquely determined by Eq. 3.4.
The following consequence of Proposition 3.2 is obvious. The following theorem is the main result of this section. It shows that a wide class of functions 0 0 has the property that the number is generator invariant. This result will also be a crucial tool in the proof of Theorem 4.4, which presents a new characterization of max-domain of attraction. The co-extremality coefficient turns out to be a measure of pairwise tail dependence, as revealed by the following result; see also Eq. 4.5. Given a D-norm on , with generator 1 , we denote by the bivariate projection of , which is itself a D-norm, with generator 1 .
Proposition 3.7 We have 0 1 , 1 , with 1 and, for , Formulated in terms of random variables, the preceding result reads as follows. Suppose the random vector 1 follows a simple max-stable df exp 1 , 0 , as in Theorem 4.1 below. Choose 1 . Then , are independent 0, and a.s. 1. The co-extremality coefficient is in this sense similar to the extremal dependence measure defined in Larsson and Resnick (2012). Another characterization of the coextremality coefficient in terms of a random vector , which is in the domain of attraction of G, is given in Eq. 4.5.
Proof We first establish part (i). Suppose that 0. Then 0 a.s. and, thus, min 0 a.s. As a consequence, max min max a.s. and thus, taking expectations, we obtain 2 max 1 1 .
Corollary 2.5 now implies 1 . If, on the other hand, 1 , then we can choose by Theorem 3.5 for in the random permutation of (2,0) with equal probability 1 2. But then 0 and, thus, 0. Next we establish part (ii). Suppose that . Then we can choose by Theorem 3.5 for in the constant random vector (1,1) and obtain 1. Now we prove the reverse direction. From Hölder's inequality we know that 0 1 2 1 2 1 with equality 1 iff a.s. But a.s. generates the bivariate D-norm .
The co-extremality coefficient behaves quite similar to the ordinary coefficient of correlation. But, while the ordinary coefficient of correlation is a measure of overall linear dependence in the data, the co-extremality coefficient addresses the upper tail of a bivariate distribution. Even more, the matrix of pairwise co-extremality coefficients turns out to be positive semidefinite, just like the ordinary correlation matrix. The following result might, therefore, enable the application of standard statistical techniques such as principal component analysis to extremal data. But this is future work. A principal component analysis for the extremal dependence measure defined in Larsson and Resnick (2012) was pursued by Cooley and Thibaud (2019).

Multivariate Extremes and D-Norms
Next we link D-norms with MEVT. A df G on is called max-stable, if for every there exists vectors 0, such that . (4.1) Recall that all operations on vectors such as addition, multiplication etc. are always meant componentwise. A df G on is a simple max-stable or simple extreme value df iff it is max-stable in the sense of Eq. 4.1, and if it has unit Fréchet margins: In this case, the norming constants are 0 and . . The theory of D-norms allows a mathematically elegant characterization of an arbitrary simple max-stable df as formulated in the next result; for a proof see Theorem 2.3.3 in Falk (2019). It comes from results found in Balkema and Resnick (1977), de Haan and , Pickands (1981), and Vatan (1985). By 1 1 1 we denote the vector in with constant entry one. This equivalent condition will ease later proofs.
Recall that a copula on is the df of a random vector 1 with the property that each follows the uniform distribution on (0,1). For an exhaustive account on copulas we refer to Nelsen (2006). Sklar's theorem (Sklar 1959(Sklar , 1996 plays a major role for the characterization of . If F is continuous, then C is uniquely determined and given by 1 1 1 1 1 0 1 where 1 inf , 0 1 , is the generalized inverse of . The copula of a random vector 1 is meant to be the copula of its df. The next result goes back to Deheuvels (1984) and Galambos (1987). It is established in Proposition 3.1.10 in Falk (2019).

Proposition 4.3 A d-variate df F satisfies
iff this is true for the univariate margins of F together with the condition that the copula of F satisfies the expansion 1 1 1 as 1, uniformly for 0 1 , where is the D-norm on that corresponds to G in the sense of Theorem 4.1.
In the next result we present a new characterization of multivariate max-domain of attraction. , which is homogeneous of order one, and is an arbitrary generator of the D-norm .
Note that by Theorem 3.5, the number does not depend on the particular generator of .
Remark 4.5 The conclusion in the preceding result is obvious for , where is an arbitrary generator of a D-norm , the random variable U follows the uniform distribution on (0,1), and U and are independent. In this case, we obtain for each measurable function 0 0 which is homogeneous of order one, 1 1 0 (use Fubini) 0 0 by Eq. 3.1. Note again that by Theorem 3.5, the number does not depend on the particular generator of . If the generator is in addition bounded, then follows a simple multivariate Generalized Pareto Distribution (GPD). In this case its df is 1 1 for large 0 . The definition of a multivariate GPD is not unique in the literature. But, if 1 is an arbitrary random vector whose copula is excursion stable and each component follows in its upper tail a standard Pareto distribution, then we call its distribution a simple GPD. The df of coincides for large 0 in this case with that of ; see Remark 3.1.3 in Falk (2019).

Proof of Theorem 4.4 The implication "
" is easily seen: Choose  The fact that 2 has a limit if t tends to infinity was already observed by (Jessen and Mikosch 2006, Section 4).
Another consequence of Theorem 4.4 is the following result, which was already established in Barbe et al. (2006). It is particularly interesting for risk assessment, as the event 1 with a large threshold t may describe an unwanted exceedance above a high threshold. By Corollary 4.6, the probability of this event is approximately as t increases, independent of the dependence structure among 1 .

Modeling Multivariate Exceedances
The following result can be used for stochastic modeling of multivariate exceedances above high thresholds and the estimation of its probability. By we denote ordinary convergence in distribution of random vectors or, equivalently, ordinary weak convergence of their corresponding distributions. For a thorough presentation of multivariate Peaks-over-Threshold modelling we refer to Rootzén et al. (2018)  If we put in particular 1 1 , where follows a copula , exp , 0 , then we obtain the above conclusion. If a multivariate exceedance above a high threshold is defined as a realization of the random vector 1 0 , with 1 1 and a large value t , then Corollary 5.2 provides its asymptotic distribution. Having independent copies 1 of , the limit distribution in Corollary 5.2 can be estimated in a straightforward manner by the ordinary empirical measure corresponding to 1 as follows. Put, for 0 and 0, According to Lemma 1.4.1 in Reiss (1993), is the ordinary empirical df of m independent and identically distributed random vectors 1 with df 1 1 1 conditional on , where is also independent of 1 . This representation allows the estimation of the distribution as n and t both tend to infinity in a straightforward manner. But details are outside the scope of this paper and require future work.
In the proof of Theorem 5.1 we make use of the following auxiliary topological result.

On Generators Whose Components Sum up to d
In this section we will see that Theorem 5.1 provides a way, how to simulate a generator 1 1 1 1 of an arbitrary D-norm on , with the additional property 1 1 1 1 , and that it delivers the distribution of 1 , which is unique by Theorem 2.3.
Take a D-norm on , choose some generator 1 of it, and let U be a random variable, which follows the uniform distribution on (0,1), and which is independent of . Put 1 .
Then the random vector 0 satisfies Choosing a large threshold 0, this equation enables the simulation of 1 in an obvious way. We will see below, in Proposition 6.2, that even the exact distribution of 1 can be derived from this equation as well.
Example 6.1 Take, for example, the Dirichlet D-norm , with parameter 0. It has the generator 1 1 1 where 1 are independent and identically gamma distributed random variables with density 1 exp , 0. It is well known that the random variables 1 1 1 1 0.
Theorem 5.1 also provides the distribution of 1 , given in the following result. This is actually one version of the angular measure, see (Falk 2019, Lemma 1.7.5), and Proposition 6.3 below. for each 0 , at which the df F is continuous. But, being df, the functions F and are both upper continuous and, therefore, for each 0 , and thus, Q is the distribution of 1 .
The arguments in the proof of Proposition 6.2 can be repeated to establish in general the distribution of a generator of a D-norm , with the particular property const, given an arbitrary norm on . The above two equations now imply the assertion.
The preceding result has the following consequence. If we know a bounded generator of a D-norm and we are able to simulate it, then we can also simulate any normed generator of with the following algorithm; recall that .
1. Sample a realization of . 2. Sample a realization u from the uniform distribution on 0 .
(a) If : Go back to step 1. (b) Else: Stop and return const .
This accept-reject algorithm runs with a random number of steps. As seen in the proof of Proposition 6.4, the probability of stopping is const for each iteration, so on average it takes const iterations to stop. Note that in the particular case 1 , we have const by the fact that 1 for each component of 1 .

On the Structure of Exchangeable D-Norms
In the paper (Mai and Scherer 2020), the authors show that the set of d-variate symmetric stable tail dependence functions, uniquely associated with exchangeable d-dimensional extreme value copulas, is a simplex and they determine its extremal boundary. It turns out that Mai and Scherer (2020) is actually a contribution to the theory of D-norms, as shown in what follows. To the best of our knowledge, we introduce the concept of an exchangeable D-norm for the first time, and we prove new results for this subfamily of D-norms. As is an arbitrary permutation of 1 , we have established exchangeability of .
Remark 7.2 One might guess that each generator of an exchangeable D-norm is exchangeable. But this is not true. Take a random variable U, which follows the uniform distribution on (0,1), and choose arbitrary numbers 0 0 In what follows we will characterize extremal exchangeable D-norms. Let 1 be a random permutation of 1 , with equal probability 1 for each possible outcome, and denote by the set of all permutations of 1 . The D-norm is obviously exchangeable. The set of D-norms is convex, see (Falk 2019, Proposition 1.4.1), and, therefore, the set of exchangeable D-norms is convex as well. It turns out that the norms , , are extremal points of it. This is the content of our next result.