On a class of norms generated by nonnegative integrable distributions

We show that any distribution function on $\mathbb{R}^d$ with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on $\mathbb{R}^{d+1}$, called $F$-norm. We characterize the set of $F$-norms and prove that pointwise convergence of a sequence of $F$-norms to an $F$-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an $F$-norm can easily be estimated by an empirical $F$-norm, whose consistency and weak convergence we establish. The concept of $F$-norms can be extended to arbitrary random vectors under suitable integrability conditions fulfilled by, for instance, normal distributions. The set of $F$-norms is endowed with a semigroup operation which, in this context, corresponds to ordinary convolution of the underlying distributions. Limiting results such as the central limit theorem can then be formulated in terms of pointwise convergence of products of $F$-norms. We conclude by showing how, using the geometry of $F$-norms, we may characterize nonnegative integrable distributions in $\mathbb{R}^d$ by simple compact sets in $\mathbb{R}^{d+1}$. We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances.


Introduction
It was observed only recently that a particular kind of norms on R d , called D-norms, are the skeleton of multivariate extreme value theory. An up-to-date account of D-norms is [4]. D-norms are de ned via a random vector (rv), called generator. The distribution function (df) of this rv, however, is not uniquely determined, and there exists an in nite number of generators of the same D-norm. It was shown by [6] that the D-norm characterizes the distribution of a generator if the constant function one is added to the generator as a further component. This led to the de nition of the max-characteristic function, which can be used to identify the distribution of any multivariate distribution with nonnegative and integrable components.
In this paper we build on these observations and construct a norm on R d+ , called F-norm, which contains the notion of max-characteristic function. In Section 2.1, we present the concept of F-norms, and show that the df of each rv X = (X , . . . , X d ) on R d with nonnegative, nonzero and integrable components can be characterized by the pertaining F-norm. We then list examples and derive basic properties as well as an inversion formula to retrieve a distribution from its associated F-norm. We also fully characterize the set of F-norms and obtain a simple classi cation in two dimensions.
In Section 3 we analyse the convergence of sequences of F-norms. We start by proving that pointwise convergence of a sequence of F-norms to an F-norm is equivalent with convergence of the pertaining dfs with respect to the Wasserstein metric. We then add some statistical views on F-norms to this section. The (random) F-norm · Fn of the empirical df Fn of a sample of n independent and identically distributed (iid) rvs is an estimator of · F with the structure of a sample mean. Local uniform consistency and asymptotic normality of · Fn as an estimator of · F are then consequences of the law of large numbers and the multivariate central limit theorem. More strongly, we establish the √ n-functional weak convergence of · Fn − · F to a Gaussian process which is essentially a functional of a Brownian bridge. Section 3 suggests that F-norms interact nicely with well-known modes of convergence and theorems of statistical analysis. In order to be able to use these norms in practice for asymptotic analyses, it is important to understand how they behave with respect to simple algebraic operations. It turns out that two F-norms can be multiplied by constructing the F-norm generated by the componentwise product of pairs of independent rvs giving rise to the individual F-norms. We also provide an integral formula making it possible, given two F-norms, to compute this product in a straightforward way. Equipped with this commutative multiplication, the set of F-norms is a semigroup with an identity element, and we can fully identify the invertible and idempotent elements for this operation. This algebraic aspect is investigated in Section 4.
The concept of F-norms as we introduce it originally focuses on multivariate rvs with nonnegative and integrable components, and thus excludes common distributions such as the multivariate normal distribution. In Section 5 we show that we can also de ne, by an exponential transformation, a concept of F-norms for a rv attaining negative values, under an integrability condition. This indeed allows us to include multivariate normal distributions, as well as other interesting examples. The multiplication of F-norms in Section 4 then represents the convolution of two rvs, and central limit theorems for iid rvs now mean pointwise convergence of the sequence of corresponding products of F-norms.
A multivariate distribution can then, under an integrability assumption, be characterized by its associated F-norm. The norm structure makes it possible to reduce the knowledge of the df F to even simpler objects than the full F-norm. Because each norm is a homogeneous function, the knowledge of an F-norm (and thus of the underlying df F) is equivalent to its knowledge on the unit simplex. Besides, and since a norm is characterized by its unit sphere, multivariate distributions on R d can be characterized, under suitable integrability conditions on the components, by the part of the unit sphere for their F-norm contained in the positive orthant of R d+ , which is a compact set. Interestingly, the convergence of F-norms, and therefore convergence of d-dimensional distributions in the Wasserstein metric, can be shown to be equivalent to the convergence of these unit spheres with respect to any Hausdor metric induced by a norm in R d+ . These geometric aspects are investigated in Section 6. Section 7 concludes.

The concept of F-norms . De nition, examples, and basic properties
Let d ≥ and X = (X , . . . , X d ) be a rv satisfying the fundamental assumption (H) Each X i is almost surely (a.s.) nonnegative with < E(X i ) < ∞.
Denote by F the df of X. For x = (x , x , . . . , x d ) ∈ R d+ , de ne a mapping · F by As shown in Theorem 2.2 below, the distribution of X is characterized by the mapping · F . This is not true in general if we replace the sup-norm · ∞ in the above de nition (1) by an arbitrary norm. Indeed, consider for instance the L -norm and de ne An arbitrary rv X ≥ ∈ R d with E(X i ) = , ≤ i ≤ d, then provides the same value x * F as the constant rv X = ( , . . . , ) ∈ R d . The use of the sup-norm · ∞ in de nition (1) is, therefore, crucial.
This paper is based on the following fundamental observations, presented in the two subsequent results. The proof of the rst result is elementary.
Moreover, the norm · F characterizes the df of X, which can be seen as follows. The function φ X , de ned for In view of the above result we call every norm on R d+ which has the representation (1) an F-norm. Let us point out that Theorem 2.2 is still valid when X is not assumed to have nonzero components, but the mapping · F is then actually only a seminorm on R d+ . Extending the de nition of the max-CF of X by considering the mapping · F thus generally leads to a seminorm rather than a norm. Observe though that unless X is the degenerate rv ∈ R d , the mapping · F induces an F-norm on R d + , where d is the number of nonzero components of X. There is therefore no loss of generality in considering F-norms rather than Fseminorms, and we do so in the remainder of this paper.
An F-norm is usually conveniently calculated by using the following fundamental formula.

Lemma 2.3.
Let F be the df of a rv X satisfying condition (H). Then, for any x = (x , x , . . . , x d ) ∈ R d+ , we have Proof. This is a straightforward consequence of the well-known formula In particular, the standard sup-norm x ∞ := max ≤i≤d |x i | on R d+ is an F-norm which characterizes the constant rv ( , . . . , ) ∈ R d .

Example 2.3 (Uniform F-norm). The uniform distribution on ( , ) is characterized by the bivariate F-norm
Example 2.4 (Exponential F-norm). The exponential distribution with mean /λ, λ > is characterized by the bivariate F-norm when x ≠ , and |x | otherwise.
We now explore some simple properties of F-norms. Each F-norm induces, as a norm, a continuous function on R d+ . It takes the value at ( , , . . . , ). It also de nes a radially symmetric function, i.e.
The norm · F is, therefore, determined by its values on [ , ∞) d+ . Additionally, any F-norm de nes a monotone norm on R d+ in the sense that These properties make it possible, in some cases, to show that certain norms are not F-norms: • the norm · := · ∞ is not an F-norm because ( , , . . . , ) = , • for any δ ∈ ( , ), the matrix M = −δ −δ is symmetric and positive de nite, and therefore induces the norm This norm is not radially symmetric, as It is actually not monotone either, since The norm · δ therefore cannot be an F-norm. The proof of the following result, which provides bounds for a general F-norm, is elementary. Proposition 2.4. Let X = (X , . . . , X d ) be a rv satisfying (H) and · F be the corresponding F-norm. For any x ∈ R d+ , we have the bounds The upper bound is always strict if both x and at least one of the x i ( ≤ i ≤ d) are nonzero.
While the upper bound in Proposition 2.4 is not an F-norm, the weighted sup-norm in the lower bound is, see Example 2.1. In the case E(X ) = · · · = E(X d ) = , this is just the standard sup-norm on R d+ . We close this section by providing results to identify those norms which are F-norms. Let us highlight rst that for any norm · on R d+ and any x ∈ R d , the function t → (t, x) is convex on [ , ∞) (and rightcontinuous at 0), and therefore automatically absolutely continuous on this interval [see e.g . 12]. With this in mind, we have the following preliminary result. Lemma 2.5. A norm · on R d+ is an F-norm if and only if the following two conditions hold: (i) it is radially symmetric, (ii) there exists a rv X = (X , . . . , X d ) which satis es (H) such that for any x , . . . , x d > , the Lebesgue In that case then · = · F with F being the df of X.
Proof. That any F-norm satis es (i) is obvious, while (ii) is a clear consequence of Lemma 2.3, reformulated as when X has df F. Conversely, let · satisfy (i) and (ii). Since · and · F are continuous, as well as radially symmetric by (i), we only need to show that x = x F for all x > . Pick such an x and write it as x = (t, /x , . . . , /x d ), for t, x , . . . , x d > . Write then, by absolute continuity, Applying Lemma 2.3 and noting that by (ii), concludes the proof.
Although the above result and in particular its part (ii) seem to be a tautology in view of the de nition of an F-norm in (1), it turns out to be quite a powerful tool of its own as illustrated by Corollaries 2.6 and 2.7 below. We start by the following simple corollary in two dimensions.

Corollary 2.6. A norm · on R is an F-norm if and only if the following two conditions hold: (i) it is radially symmetric, (ii) the Lebesgue derivative of t → (t, ) is almost everywhere equal to a univariate df F on [ , ∞) with a nite rst moment equal to ( , ) .
In that case then · = · F .
which does not de ne a df on [ , ∞) having a (strictly) positive rst moment.
Indeed, it is clearly radially symmetric and which de nes the df of a Burr type III distribution in the sense of [1, Table 2.1]. This distribution, for p > , has a nite rst moment.
Even though providing a simple characterization of F-norms in arbitrary dimensions appears to be a di cult problem due to the high-level condition (ii) in Lemma 2.5, there is a simple inversion formula inspired by this result that makes it possible to go from an F-norm to its pertaining df. This is the focus of the following result, which can also be used to check that a norm is not an F-norm. Its proof is a straightforward consequence of Lemma 2.3 and right-continuity of the df F. The fact that the df F is determined by · F is another obvious consequence.
which de nes the df of the degenerate vector ( , . . . , ). This distribution does not have strictly positive marginal moments and thus, by Corollary 2.7, · cannot be an F-norm.

Limiting behavior and estimation of F-norms
While the pointwise limit of a convergent sequence of D-norms is again a D-norm [see 4, Corollary 1.8.5], this is not true for F-norms: for instance, if (pn) is a sequence of real numbers with pn > and pn ↓ , then · pn → · , and · pn is for each n an F-norm, but the limit · is not. However, if we ask that the limit is an F-norm, then we can relate the convergence of F-norms with convergence of distributions in the Wasserstein metric. Recall that the Wasserstein metric between two probability distributions P, Q on R d with nite rst moments in each component is Convergence of probability measures Pn to P on R d with respect to the Wasserstein metric is equivalent to weak convergence together with convergence of the moments see e.g. [16, De nition 6.8 and Theorem 6.9]. With this de nition in mind, we can show the following result.

Theorem 3.1. Pointwise convergence of a sequence of F-norms · Fn to an F-norm · F is equivalent to convergence of the sequence of distributions Fn to F in the Wasserstein metric.
Proof. Pointwise convergence of · Fn to · F implies pointwise convergence of the sequence of max-CFs of Fn (as de ned in (2)) to the max-CF of F, which entails the desired convergence in the Wasserstein metric by Theorem 2.1 in [6].
Conversely, if Fn → F in the Wasserstein metric, let X (n) and X have dfs Fn and F. For any An analogue inequality holds if we switch X (n) and X. We can then integrate to nd Since X (n) and X were arbitrary rvs having dfs Fn and F, this yields which concludes the proof.
The nice behavior of F-norms with respect to sequences of distributions naturally raises the question of what happens when Fn is chosen to be the empirical df based on iid copies X ( ) , . . . , X (n) of a rv X satisfying (H), i.e.
The (random) F-norm generated by Fn is nothing but The law of large numbers then implies, for each x ∈ R d+ , that a.s.
This convergence suggests that the estimation of an F-norm is completely straightforward; by contrast, estimating the related concept of a D-norm in the context of multivariate extreme value analysis requires quite sophisticated techniques. We now provide further insight into the convergence of · Fn to · F . Noting that for any the following locally uniform re nement of the pointwise almost sure convergence of · Fn to · F is a direct consequence of the continuity of · F . Theorem 3.2. Let X ( ) , . . . , X (n) be iid copies of a rv X satisfying (H), with df F. Let · Fn be the random F-norm generated by the empirical df Fn of this sample. We then have, for any x ≥ , To analyse the rate of (uniform) convergence of · Fn to · F , we de ne the empirical F-norm process This stochastic process has continuous sample paths and satis es Sn( ) = . Suppose then that E(X i ) < ∞ for any i ∈ { , . . . , d}. Based on the standard central limit theorem, which gives the pointwise asymptotic normality of Sn, we may ask the question of the limiting behavior of the process Sn. For ease of exposition, we state a result in the case d = .
Theorem 3.3. Let X ( ) , . . . , X (n) be iid copies of a univariate rv X with df F. Assume that X is nonnegative, with nonzero expectation and nite variance. Let · Fn be the random F-norm generated by the empirical df Fn of this sample. For any x , y > , we have where the limiting process S, which should be read as 0 when y = , is a bivariate Gaussian process with covariance structure Under the further assumption that /π, and thus du is well-de ned and nite with probability 1. It is then straightforward to show, using the covariance properties of W, that the covariance structure of this Gaussian process coincides with that of S.
Proof. The random functions Sn and S are elements of the functional space C([ , x ] × [ , y ]). By Theorem 7.5 in [2], it su ces to show the convergence of nite-dimensional margins of Sn to those of S along with tightness of (Sn), in the sense of tightness of its sequence of distributions.
We start by convergence of nite-dimensional margins. The multivariate central limit theorem implies, for nonnegative pairs (x , y ), . . . , (x k , y k ), that the rv (Sn(x , y ), . . . , Sn(x k , y k )) converges weakly to a centered Gaussian distribution. By Hoe ding's identity [see 4, Lemma 2.5.2], the limiting covariance matrix is described by This is clearly equal to 0 when either y i or y j is 0, and otherwise, using the change of variables x = x i u, y = x j v, we nd which is exactly the covariance structure of the Gaussian process S. We now show tightness, that is, for any ε > , or, in other words, that Sn is stochastically equicontinuous on [ , x ] × [ , y ]. The key to the proof is threefold. Firstly, we apply Theorem 1 p.93 of [15] to construct, on a common probability space, a triangular array (U (n, ) , . . . , U (n,n) ) n≥ of rowwise independent, standard uniform rvs, and a Brownian bridge W such that Since Using the identity max(a, b) = a + b − min (a, b), valid for any a, b ≥ , it follows that: with Tn(x, y) := y The rst term on the right-hand side above is stochastically equicontinuous, because the random term is a O P ( ) (by the Chebyshev inequality). We conclude the proof by focusing on Tn(x, y), and for this we rst remark that almost surely. A consequence of this convergence is that, to show the stochastic equicontinuity of Tn, it is enough to prove that the random function T de ned by Recall that W has almost surely continuous sample paths on This completes the proof.
In the case d > , and under regularity conditions [e.g. those of 10], a similar proof using a special construction of the multivariate empirical process can be written to show an analogue of Theorem 3.3, which gives the convergence of the process Sn, in a space of continuous functions over compact subsets of [ , ∞) d+ , to a (d + )−dimensional Gaussian process S with covariance structure Our objective is now to dwell upon the nice sequential behavior of F-norms and show an example of how this could be used to prove powerful theorems on the convergence of certain sequences of rvs. To this end we rst need to understand better how to manipulate F-norms, which leads us to exploring their algebraic properties.

Algebra of the set of F-norms
One can multiply F-norms · F and · G by constructing the F-norm generated by the componentwise product of any pair of independent rvs having dfs F and G; independence is used to ensure that the distribution of this componentwise product is well-de ned, and thus so is the product F-norm. We denote this operation by · F * · G . It coincides with taking products of D-norms if · F and · G have components with expectation 1, see [4,Section 1.9]. In general, the product F-norm can be calculated using the following Tonelli formula. , x t , . . . , x d t d ) G dF(t , . . . , t d ).

Proposition 4.1. Let F and G be the dfs of two rvs satisfying condition (H). Then, for any
Proof. Let X = (X , . . . , X d ) and Y = (Y , . . . , Y d ) be independent and have dfs F and G. We have By nonnegativity of max (|x | , |x | X Y , . . . , |x d | X d Y d ) and independence of X and Y, we nd, using the Tonelli theorem, that which is exactly the rst formula. The second expression follows by swapping integration with respect to dG for integration with respect to dF.

Example 4.1 (Product of uniform F-norms). Following Example 2.3, the product of the standard uniform Fnorm by itself has the expression
Let us now explore in more detail the structure of the set of F-norms equipped with its multiplication. It is clear that the sup-norm · ∞ on R d+ , with generator ( , . . . , ) ∈ R d , is an identity element for this operation. It is also straightforward to see that it is the unique such element: if · F is an identity element for * then · F = · F * · ∞ = · ∞ . We summarize this short discussion by the following result.

Proposition 4.2. The set of F-norms is a commutative monoid for the F-norm multiplication *, with identity element · ∞ . The only invertible elements are the F-norms generated by nonrandom vectors.
The only point we need to show in Proposition 4.2 is the assertion about invertible elements. The key is to note the following lemmas.

Lemma 4.3. Let Z be a real-valued rv such that |E(e itZ )| = for any t ∈ R. Then Z is almost surely constant.
Proof of Lemma 4.3. We use the Cauchy-Schwarz inequality for the inner product (X, Y) → E(XY) on the space of complex-valued square-integrable rvs, to obtain: ∀t ∈ R, |E(e itZ )| = |E(e itZ · )| ≤ .
By assumption, we actually have equality here. This means that for any t, the rvs e itZ and 1 are almost surely proportional, i.e. e itZ = λ(t), with λ(t) ∈ C. De ne now the event E t := {e itZ = λ(t)}, and let (tn), (t n ) be two sequences converging to 0. De ne E = ( n E tn ) ∩ ( n E t n ). Then P(E) = and on E, It follows that the limit lim n→∞ λ(tn) − λ( ) tn exists and does not depend on the choice of tn → : the function λ is di erentiable at 0. Conclude, by using (tn) again, that on the event ( n E tn ), λ ( ) = iZ and thus Z is almost surely the constant −iλ ( ). Proof of Proposition 4.2. Let · F and · G satisfy · F * · G = · ∞ . Equivalently, there are independent rvs (X , . . . , By Theorem 2.2, we nd that each X i Y i is a.s. constant equal to 1. Conclude by applying Lemma 4.4.
The same kind of argument can be used to identify the set of idempotent elements for the multiplication of F-norms.

Proposition 4.5. The only idempotent element for multiplication of F-norms is the sup-norm · ∞ .
The proof is again based on an auxiliary result for real-valued rvs. Proof of Lemma 4.6. The assumption is P(XY ≤ t) = P(X ≤ t) for any t. Note that so that P(X = ) ∈ { , }, and necessarily P(X = ) = since E(X) > . Then by assumption log X + log Y d = log X. If φ(t) := E(e it log X ) denotes the characteristic function of log X, this entails [φ(t)] = φ(t) for any t ∈ R, by independence. Thus, for any t ∈ R, φ(t) ∈ { , }. Noting that φ( ) = and φ is continuous entails that necessarily φ ≡ , since φ(R) must be a path-connected subset of { , }. As a consequence, log X = almost surely, completing the proof.
Proof of Proposition 4.5. Let · F be an idempotent element for the multiplication of F-norms, with generator (X , . . . , X d ). Let (Y , . . . , Y d ) be an independent copy of this rv. Since · F is idempotent, we have, for any

F-norms of general random vectors
The concept of F-norms focuses on the distribution of an arbitrary multivariate rv with nonnegative and integrable components. Our purpose here is to show how we can also de ne, in a sensible way, a concept of F-norms for a rv whose components can attain negative values, under an integrability condition.
Let X = (X , . . . , X d ) be an arbitrary rv satisfying E(exp(X i )) < ∞, ≤ i ≤ d. Then Y := exp(X) = (exp(X ), . . . , exp(X d )) generates an F-norm · exp F . As the function x → exp(x) is a bijection from the real line onto the interval ( , ∞), the distribution of X is characterized by the F-norm · exp F , which we call a log F-norm.
Example 5.1 (Normal distribution). Put Z := exp(X − σ / ), where X follows the univariate normal distribution N( , σ ). The rv Z is log-normal distributed with E(Z) = . The log F-norm of X − σ / is then just a D-norm and equals, for x, y > , which is the so-called Hüsler-Reiss D-norm with parameter σ > [see 4]; by Φ we denote the df of the standard normal distribution on R. As a consequence, the normal distribution N(−σ / , σ ) of X − σ / is characterized by the preceding norm · exp F . More generally, the log F-norm of the normal distribution N(µ, σ ) with arbitrary µ ∈ R and σ > is, for x, y > , By Corollary 2.6, we should nd back the log-normal df from this F-norm by di erentiating (t, ) exp F on ( , ∞). Clearly Note also that to nd, as expected: Combining the discussion we have developed in the previous example with Theorem 3.1 leads, without any further calculation, to the following immediate result. This serves as a further example of how the asymptotic results in Section 3 may be used to establish asymptotic theory.
Corollary 5.1. Let (µn), (σn) be real-valued sequences such that µn → µ and σn → σ > . Then: • The sequence of log-normal distributions with parameters µn and σ n converges to the log-normal distribution with parameters µ and σ in the Wasserstein metric.

• The sequence Gn of normal distributions with parameters µn and σ n converges in distribution to the normal distribution G with parameters µ and σ , and the moments of Gn converge to those of G.
More generally, if X follows a multivariate normal distribution N(µ, Σ) with mean vector µ ∈ R d and covariance matrix Σ = (σ ij ) ≤i,j≤d , then each component Y i = exp(X i ) is log-normal distributed with mean E(Y i ) = exp(µ i + σ ii / ). In analogy to the D-norm generated by the normalized rv Z = Y /E(Y) and called a Hüsler-Reiss D-norm [see 4], we call the F-norm corresponding to Y a Hüsler-Reiss F-norm. It characterizes the normal distribution N(µ, Σ).
The concept of log F-norms for rvs with an arbitrary sign is not adapted solely to Gaussian distributions, as we show in the following examples.
Example 5.2 (Gumbel distribution). Let X have the standard negative Gumbel distribution, i.e.
Then exp(X) has a unit exponential distribution, and therefore the log F-norm characterizing the standard negative Gumbel distribution is when x ≠ , and |x | otherwise (see Example 2.4).

Example 5.3 (On the central limit theorem)
. Let X ( ) , X ( ) , . . . be iid copies of a centered rv X = (X , . . . , X d ) having covariance matrix Σ, and a nite moment generating function in a neighborhood of the origin, i.e. there exists ε > with φ j (t) := E(exp(tX j )) < ∞ for any |t| < ε and ≤ j ≤ d. The multivariate central limit theorem and continuous mapping theorem imply where ξ = (ξ , . . . , ξ d ) follows a multivariate normal distribution with mean vector zero and covariance matrix Σ. Besides, we have Since X j is centered with variance Σ jj , we have by a Taylor expansion It follows that the sequence has a bounded second moment and thus is uniformly integrable [see e.g. 2] for each j = , . . . , d. This entails convergence of the sequence of its rst moments and, combined with (4) and Theorem 3.1, pointwise convergence of the generated log F-norms, i.e.
for each x , x , . . . , x d ≥ . We thus have a convergence of F-norms akin to the central limit theorem.
We could, of course, have used in place of the exponential function any one-to-one increasing transformation from R to ( , ∞) in order to de ne an F-norm for general rvs. The exponential function, however, interacts well with our notion of product of F-norms, in the sense that · exp X * · exp Y = · exp X+Y if X and Y are independent: a product of two log F-norms is the log F-norm corresponding to the convolution of their individual distributions.

Geometry of F-norms
The original motivation for constructing F-norms was to combine the distributional properties of a max-CF with the structure of a D-norm into a single mathematical object. We have so far concentrated on the information that F-norms bring about multivariate distributions. We use here the geometry of the F-norms to nd yet other di erent objects who summarize a multivariate distribution.
Since any norm · on R d+ is homogeneous, an immediate consequence is that each F-norm · F is uniquely determined by its values on the unit sphere for · , namely S · := u ∈ R d+ : u = : to put it di erently, we have for x ∈ R d+ , x ≠ , with x/ x ∈ S · . By choosing · = · and using the radial symmetry of the L -norm and of F-norms, we nd that we need only consider the values of · F on the part of the sphere S · contained in [ , ∞) d+ . In other words, each df F of a rv X satisfying (H) is characterized by the function , and we therefore call the function A the Pickands dependence function of the F-norm · F . Let us brie y mention here that, based on a sample of copies of X, we can estimate this Pickands dependence function by an empirical version, just as we did in Section 3 for the full F-norm: let X ( ) , . . . , X (n) be iid copies of a rv X satisfying (H). Put, for t ∈ ∆ , with t := − d i= t i , which is that (random) Pickands dependence function which characterizes the empirical df Fn. The asymptotic properties of An follow directly from our asymptotic results in Section 3: since ∆ is compact, we get, by weakly in the space of the continuous functions on the unit simplex in R d+ , where S is a Gaussian process. We now explore how, instead of characterizing an F-norm by a function such as its Pickands dependence function, we can identify it by a compact set which summarizes the geometry of an F-norm. Recall that an F-norm is characterized by its values on any sphere S · , where · is an arbitrary norm on R d+ . By choosing · = · F and using the radial symmetry of any F-norm, we obtain the following corollary.
This corollary provides a compact set characterizing any multivariate distribution with nonnegative, nonzero and integrable components. For such distributions, it is therefore an alternative to the lift zonoid studied by [9] and [11]. The next two examples show how this set can be computed in practice.
Example 6.1 (Unit sphere for the uniform F-norm). Let F be the uniform distribution on ( , ). We know from Example 2.3 that this distribution is characterized by the bivariate F-norm given by As a consequence, the set S + · F corresponding to this norm is the set This set is represented in Figure 1. . If x, y > are such that (x, y) ∈ S + which implies, if λ := y/x ∈ ( , ∞), that It is readily checked that conversely, any point of the form , so that we have a parametrization of S + · F making it possible to represent this set. This is done in Figure 2 for various values of σ. One can observe in this Figure that, as should be apparent from the parametrization, the limit σ ↓ produces the part of the sphere of the sup-norm on R contained in the upper right quadrant, while the limit σ → ∞ yields the segment {(x, − x), ≤ x ≤ }, corresponding to the sphere of the L −norm. Example 6.2 suggests that the convergence of F-norms, and thus convergence of the pertaining distributions in the Wasserstein metric, is at least informally linked to the convergence of their unit spheres. To make this intuition rigorous, we recall the de nition of a Hausdor metric. If · is an arbitrary norm on R d+ and A, B are two subsets of R d+ , we let their · -Hausdor distance to be Intuitively, two sets A and B are therefore close in the · -Hausdor metric if and only if each point in A (resp. B) is close, in terms of · , to at least one point in B (resp. A). Such a distance may be in nite under no further assumptions on A and B, but is always nite if A and B are bounded. With this de nition in mind, we have the following result. j → E X j as n → ∞, and thus, since X (n) , X satisfy (H), there is c > such that E X j ≥ c and E(X (n) j ) ≥ c for any n. De ne then a weighted sup-norm · ∞,c on R d+ by (x , x , . . . , x d ) ∞,c := max(|x |, c|x |, . . . , c|x d |).
By Proposition 2.4, we obtain · F ≥ · ∞,c and · Fn ≥ · ∞,c for any n. Consequently, if B denotes the closed unit ball for the norm · ∞,c and B := B∩[ , ∞) d+ , then B contains S + · F and the S + · Fn for any n. In addition, by inequality (3) and since B is compact, Assume from now on that n is so large that un < . Pick x in S + · F . Then since B contains S + · F , we have This also entails x Fn ≥ − un. Note then that x/ x Fn ∈ S + · Fn and thus , Write then x/ x F ∈ S + · F , which yields From (6) and (7)  . Note that x/ x Fn ∈ S + · F n and thus, by assumption, there is a sequence (z (n) ) ⊂ S + By the reverse triangle inequality, this entails This shows that / x Fn → and thus x Fn → = x F as required.

Conclusion
D-norms are tailor-made for multivariate extreme value theory as they turn out to provide an easily accessible common thread, in the sense that they do not require the knowledge of multivariate regular variation and of the associated topology background. Our paper introduces F-norms, which are an o spring of D-norms, to address the general framework of multivariate distributions rather than the max-stable distributions that are the focus of multivariate extreme value theory. While there is currently no competitor to the concept of D-norms in multivariate extreme value theory, there are of course various competitors to the concept of Fnorms, such as the Fourier and Laplace transforms. As this paper shows, F-norms have their place in the probabilistic toolbox; while the pointwise convergence of Fourier and Laplace transforms is linked to weak convergence, the convergence of a pointwise sequence of F-norms translates into Wasserstein convergence of the underlying sequence of distributions. In this sense, F-norms behave like the max-characteristic function of [6], but their added norm structure o ers an interesting geometric characterization of a distribution by a compact set, in a di erent way to existing alternatives such as the lift zonoid. As a corollary, the use of Fnorms provides a nice interpretation of convergence in the Wasserstein metric by the means of convergence in Hausdor metrics. A promising aspect of F-norms is the good behavior of their random sample counterparts, as illustrated in Section 3, and the associated consequences this may have for statistical methodology. For instance, it is not di cult to imagine a goodness-of-t test based on Theorem 3.3. The generalization of the results of Section 3 on the convergence of empirical F-norms to the case of stationary but dependent data is another interesting problem. Theorem 3.2 rests on the strong law of large numbers, which is known to be true for stationary and, say, mixing sequences under appropriate conditions, see for instance [14] in the context of ρ−mixing. Generalizing Theorem 3.3 would require the use of empirical processes techniques for dependent data, for which a good starting point is [3]. These lines of investigation are beyond the scope of the paper and will be part of future research on the topic of F-norms.
(reference 41710) is gratefully acknowledged. Both authors are indebted to two anonymous reviewers for their careful reading of the manuscript and their constructive remarks.