Multivariate extreme value theory

Johan Segers; Philippe Naveau

REVIEW 2 major objections 4 minor 1 cited by

Multivariate extreme value theory

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This chapter shows that multivariate extreme value distributions are exactly the max-stable distributions, with all three standard representations governed by one exponent measure.

desk verdict Useful introductory exposition of multivariate EVT from the MGP viewpoint, but Example 5.1's logistic generator range α>1 is internally inconsistent and needs fixing. read the letter →

arxiv 2412.18477 v1 pith:5MBEJX3A submitted 2024-12-24 math.ST stat.TH

Philippe Naveau , Johan Segers This is my paper · ORCID

classification math.STstat.TH MSC 60G7062G3260G55

keywords multivariateextremevaluetheorygeneralizedParetodistributionsmax-stableexponentmeasurepointprocessestaildependenceangularstablefunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This chapter sets out to give a single conceptual architecture for extremes in two or more dimensions. Its thesis is that three classical-looking constructions—multivariate generalized Pareto (MGP) distributions for excesses over high thresholds, Poisson point processes driven by an exponent measure, and max-stable multivariate extreme value (MEV) distributions for componentwise maxima—are equivalent portraits of one object, the tail dependence structure. The route starts with the most permissive definition of a multivariate exceedance (at least one coordinate above its threshold), builds the MGP family from a unit-exponential common shock, and then shows how the exponent measure is the intensity connecting this family to point-process limits and to the formula $G(x)=\exp[-\ell\{-\log G_1(x_1),\ldots,-\log G_D(x_D)\}]$. The chapter matters because extremal dependence has no finite-dimensional analogue of the correlation matrix; this framework explains why and supplies the nonparametric substitute.

What carries the argument

The load-bearing object is the exponent measure $\Lambda$, a measure on $[-\infty,\infty)^D\setminus\{-\infty\}$ with unit-exponential scale, defined by the normalization $\Lambda(\{x:x_j\geq 0\})=1$ for each coordinate and the homogeneity property $\Lambda(B+t)=e^{-t}\Lambda(B)$. It enters through the limiting relation $P(E\in B+t)\sim e^{-t}\Lambda(B)$ for a vector $E$ with unit-exponential margins, which yields Poisson convergence of the counting variables $N_n(B)$ by the law of small numbers. From $\Lambda$ one obtains both the stable tail dependence function $\ell(y)=\Lambda(\{x:x\not\leq \log y\})$ and the standard MGP distribution via $P(Z\in B)=\Lambda(B)/\Lambda(L)$, while the angular measure on the unit simplex gives a geometric way to model $\Lambda$. The three views are different ways to write down the same measure.

What would settle it

For a bivariate vector $E$ with unit-exponential margins and a Clayton copula with $\theta=1$, compute the Poisson expectation in Eq. (25) for $B=\{x:x_1>0, x_2>0\}$: it is $n(2e^{\log n}-1)^{-1}\to 1/2$, matching an exponent measure with interior mass $1/2$. Now try the same calculation for a valid copula whose ratio $P(E\in B+t)/e^{-t}$ oscillates between two positive values as $t\to\infty$; any such copula would make Eq. (21) fail, so no exponent measure would exist for it and the three-way equivalence would not apply. This is the specific calculation that identifies which distributions lie inside the framework and which lie outside it.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is an equivalence chain centered on the exponent measure. Proposition 4.1 says that a D-variate distribution with non-degenerate margins is a multivariate extreme value distribution if and only if it is max-stable, and whenever this holds its distribution function can be written as $G(x)=\exp[-\ell\{-\log G_1(x_1),\ldots,-\log G_D(x_D)\}]$, where $\ell$ is the stable tail dependence function derived from the exponent measure. The same exponent measure, normalized by its mass on the L-shaped set $L=\{x:x\not\leq 0\}$, produces the standard multivariate generalized Pareto vector $Z$ through $P(Z\in B)=\Lambda(B)/\Lambda(L)$ (Proposition 3.1), so the MGP distribution and the point-process intensity view are mutually recoverable. The chapter also stresses that the equivalence is nonparametric: no finite-dimensional family can capture all possible tail dependence structures, making the exponent measure or its angular measure the natural infinite-dimensional object.

Load-bearing premise

The construction rests on the assumption that the joint tails of the standardized vector settle into a stable exponential decay with a fixed proportionality constant as the thresholds rise; if this multivariate domain-of-attraction condition fails, the exponent measure, the Poisson point-process limit, and the MEV limit in Theorem 4.2 are not defined.

Editorial extensions

If this is right

Any model fitted in one of the three representations can be translated into the other two: a fitted MGP distribution determines the exponent measure and hence the MEV distribution, and conversely.
The extremal coefficient $\Lambda(L)=\ell(1)\in[1,D]$ gives a one-number summary of tail dependence, equal to 1 in complete dependence, to $D$ in asymptotic independence, and in dimension two related to the tail dependence coefficient by $\Lambda(L)=2-\chi$.
The threshold-stability propositions imply that multivariate peaks-over-threshold models remain self-consistent when the threshold is raised, the direct multivariate analogue of the univariate generalized Pareto stability property.
The max-stability characterization means a candidate distribution for block maxima is valid exactly when it can be written through a stable tail dependence function with GEV margins; no additional parametric structure is needed.
For nonnegative linear combinations with a common shape parameter, a positive combination of an MGP vector is univariate generalized Pareto with scale equal to the weighted sum of the component scales, independently of the dependence structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Implicit in the equivalence is a transfer principle: estimation and model checking can be carried out in whichever representation is most convenient, and the implied extremal coefficient should agree across representations; comparing an MGP-based estimate with a block-maxima-based estimate on the same data would test the framework end to end.
Because the asymptotic-independence boundary appears as zero mass of the exponent measure on the interior of the L-shaped set, a natural extension is to use hidden regular variation to zoom into the slower decay at that boundary, a direction the chapter explicitly leaves to other chapters.
The representation $\ell(y)=E[\max(y e^U)]$ reads tail dependence as a maximum over independent shocks, which suggests generative models in which $U$ is driven by covariates or spatial fields; the equivalence guarantees that any such generative specification automatically yields a valid MGP and MEV model.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I've read the Naveau–Segers chapter. The short version: this is a well-crafted teaching chapter, not a research paper, and its one substantive flaw is a concrete error in Example 5.1.

What it does well: it organizes multivariate EVT around MGP distributions and the unit-exponential scale, which genuinely makes the additive formulas cleaner. The failure-set/point-process perspective is explained clearly, and Section 7 contains an actual proof of Proposition 3.1 rather than just citing it. The "cheat-sheet" Table 1 comparing MGP to the Gaussian case is a nice touch. Propositions 2.2–2.5 are quoted from Rootzén–Segers–Wadsworth, and Proposition 4.1/Theorem 4.2 are classical; the chapter says so. The self-citations are to parameter-free derivations with stated assumptions; I don't see circularity.

The weak spot: Example 5.1 as written has no admissible parameter value. The displayed pU has Γ(1-1/α), which requires α>1 for E(e^{U_j}) to be finite and positive. But the displayed logistic ℓ(z) has extremal coefficient ℓ(1)=D^α, and for α>1 that exceeds D, violating the bound (32). Equivalently, ℓ is not convex for α>1. So the generator cannot produce the claimed ℓ for any α. This is not a cosmetic slip; it's the example that shows how U-generators connect to named MEV models. The fix is standard: the valid range for the logistic ℓ is α∈(0,1], and the U-generator needs a different parametrization or a different stated relationship. Everything else in the chapter is unaffected.

Minor: Eq (21) is stated for 'not too rough' sets and the domain-of-attraction condition is not formalized. For an introductory chapter that's acceptable, though a footnote would help.

Who it's for: non-specialists, graduate students, and practitioners who want a readable map of the MGP/exponent-measure/max-stable landscape. It is not a research contribution and shouldn't be reviewed as one, but as a handbook chapter it deserves competent refereeing — the Example 5.1 error needs correcting before it goes to press.

Referee Report

2 major / 4 minor

Summary. This expository chapter develops multivariate extreme value theory from the perspective of multivariate generalized Pareto (MGP) distributions, then connects that perspective to exponent measures, Poisson point processes, and multivariate extreme value (MEV) distributions obtained as limits of componentwise maxima. The organizing thread is the use of failure sets and the unit-exponential scale: threshold exceedances are described by MGP vectors, the Poisson limit of counting variables yields the exponent measure, and the MEV representation G(x)=exp[-l{-log G1(x1),...,-log GD(xD)}] follows from the absence of points in a shifted risk region. The chapter also contains parametric examples (logistic, Hüsler-Reiss, T-Gaussian), a set of stability properties, a comparison table with the Gaussian case, and a proof of Proposition 3.1 in the complements section. The core equivalences among max-stability, exponent measures, and the MEV representation are classical and are presented correctly, but the logistic example in Section 5 has a genuine parameter-range inconsistency that needs to be fixed.

Significance. If the local defect in Example 5.1 is corrected, this would be a valuable pedagogical synthesis: it gives a unified, mostly self-contained route from MGP threshold exceedances through point processes to MEV distributions, with explicit density formulas (16)-(17), threshold-stability propositions, and a useful cheat-sheet in Table 1. The main theoretical claims, especially Proposition 4.1 and the representation in Eq. (53), are standard and sound; I found no internal inconsistency in the abstract max-stability/exponent-measure framework. The chapter also gives credit where it is due by explicitly connecting its U- and T-generator constructions to earlier work, and it is honest about deferred proofs. The load-bearing weakness is Example 5.1, where the stated generator density and the stated logistic stable tail dependence function have no common admissible parameter value; since Section 5 is the part that makes the generator-to-MEV bridge concrete, this must be repaired before publication.

major comments (2)

[Section 5, Example 5.1] The displayed pair (pU, l) is internally inconsistent. The stable tail dependence function l(z)=(z1^{1/alpha}+...+zD^{1/alpha})^alpha is valid only for alpha in (0,1]: for alpha>1, l(1)=D^alpha>D, violating the extremal-coefficient bound in Eq. (32), and l is not convex (for D=2 and alpha=2, the Hessian at (1,1) has eigenvalues 0 and -1, so the function is not a stable tail dependence function). But the stated U-density forces alpha>1: the moment condition E(e^{Uj})=Gamma(1-1/alpha) is possible only when 1-1/alpha>0, i.e., alpha>1, because E(e^{Uj}) must be positive and finite; for alpha in (0,1], Gamma(1-1/alpha) is negative or has poles, and already in dimension D=2 the normalizing constant alpha^{D-1}Gamma(D-1/alpha)/Gamma(1-1/alpha) equals alpha-1<0 for alpha in (0,1). Thus the example as written has no admissible parameter value that simultaneously supplies a valid generator density and the claimed logistic MEV model. Please correct the parameter range or the density and its normalizing constant, and also align the U standardization with Eq. (51), which requires E(e^{Uj})=1 rather than E(e^{Uj})=Gamma(1-1/alpha).
[Section 3, Eq. (21) and Section 4, Theorem 4.2] The derivation of the Poisson limit for Nn(B) and of the MEV limit in Theorem 4.2 rests on the existence of the exponent measure Lambda in Eq. (21). The chapter states this informally as holding 'in many cases' and does not spell out the underlying condition. I recommend adding one sentence making explicit that Eq. (21) is a multivariate regular-variation/domain-of-attraction condition and that without it the limits in Eqs. (25)-(26) and the convergence in Theorem 4.2 need not hold. This is not a mathematical error in the current text, because Section 4 explicitly assumes Eq. (21) when deriving the limit, but the condition is load-bearing and deserves to be stated as an assumption rather than as a generic property.

minor comments (4)

[Section 7, proof of Proposition 3.1] In the proof, the set A is written as A subset [−∞,−∞)^D; this should presumably be A subset [−∞,0]^D (or similar), since S=Z−E takes values in [−∞,0]^D.
[Section 4, Theorem 4.2] Theorem 4.2 is stated without proof or citation, while the only proof in Section 7 is that of Proposition 3.1; adding a reference to a standard textbook treatment of multivariate maxima convergence (e.g., [8, Chapter 6] or [17]) would help the reader.
[Section 3, Eq. (22)] Equation (22) uses the non-strict inequality x_j>=0, whereas the surrounding discussion and Eq. (20) use strict inequalities such as x_j>u; please align these conventions or add a sentence explaining that the relevant boundaries are Lambda-null.
[Section 5, Example 5.1] Even after correcting the parameter range, the displayed generator density should be checked against the general theory: the sentence 'the choice of U such that each component satisfies E{exp(Uj)} = Gamma(1-1/alpha)' is not compatible with the standing requirement E(e^{Uj})=1 in Eq. (51), so the standardization step needs to be made explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the MGP/exponent-measure/MEV equivalences are classical or proved from stated definitions, and the self-citations are supporting facts rather than premises that force the conclusions.

full rationale

The derivation chain is self-contained and non-circular. Proposition 3.1 is proved in Section 7 directly from Definition 2.1 and Definition 3.1, using only homogeneity, max S = 0, and S <= 0; the two directions establish the bijection between exponent measures and standard MGP vectors rather than presupposing it. Definition 4.1 introduces MEV distributions via the stable tail dependence function, Proposition 4.1 states the classical max-stability characterization as a theorem, and Theorem 4.2 derives the MEV limit from the Poisson convergence in Eq. (26) under the explicitly stated domain-of-attraction condition in Eq. (21), which is an assumption about the input distribution rather than a restatement of the target result. The citations to the authors' prior work [12,18] are used for supporting distributional identities, threshold-stability facts, and parametric examples; these cited results are parameter-free, have stated assumptions, and are not needed as premises to force the central equivalence. The one notable defect is a correctness inconsistency, not a circularity: in Example 5.1 the displayed logistic ℓ with α > 1 violates the extremal-coefficient bound ℓ(1) <= D in Eq. (32), while the Γ(1 − 1/α) moment condition excludes the valid α ∈ (0,1] range; nothing in that example assumes the claimed conclusion as an input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new entities are postulated. The common shock E and deviation vector S in Definition 2.1 are mathematical representations of known MGP distributions, not new physical or probabilistic objects. No free parameters are fitted to data; the model parameters in Section 5 (alpha, mu, Sigma) are generic parameters of standard families, not fitted to any dataset.

assumptions (4)

domain assumption The distribution of E with unit-exponential margins satisfies the limit (21) for some exponent measure Lambda, i.e., it is in the max-domain of attraction of a multivariate extreme value distribution.
Invoked in Section 3, Eq (21), to define Lambda and to derive Poisson convergence of the counting variables (24). All later point-process and MEV results rely on this condition.
standard math Univariate extreme value limit theorems: GEV distributions are the only nondegenerate limits of normalized maxima and GP distributions are the only limits of threshold exceedances.
Used as background in Sections 1, 2, and 4, notably in Theorem 4.2 where the marginal convergence is assumed from univariate theory.
standard math Convergence in distribution of the counting processes N_n in (24) to a Poisson point process with intensity measure Lambda holds for sets B with Lambda-null boundary.
Stated in Section 3 via the law of small numbers; the chapter explicitly says the formal theory goes beyond its scope, so this is an imported background result.
standard math Conventions on the extended real line, including points with coordinates in [-infinity, infinity) and the convention 0 times (-infinity) = 0 in linear transformations.
Used throughout Section 2 and in Proposition 2.5 to handle -infinity coordinates in S.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Multivariate extreme value theory." pith.science (2026). https://pith.science/paper/5MBEJX3A

@misc{pith2026241218477,
  author       = {Pith},
  title        = {Pith review of: Multivariate extreme value theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MBEJX3A}},
  note         = {Machine review of arXiv:2412.18477}
}

read the original abstract

When passing from the univariate to the multivariate setting, modelling extremes becomes much more intricate. In this introductory exposition, classical multivariate extreme value theory is presented from the point of view of multivariate excesses over high thresholds as modelled by the family of multivariate generalized Pareto distributions. The formulation in terms of failure sets in the sample space intersecting the sample cloud leads to the over-arching perspective of point processes. Max-stable or generalized extreme value distributions are finally obtained as limits of vectors of componentwise maxima by considering the event that a certain region of the sample space does not contain any observation.

Figures

Figures reproduced from arXiv: 2412.18477 by the authors.

**Figure 1.** The pair M = (M1, M2) T of componentwise maxima of a sample of observations in the plane is dominated by a threshold vector u = (u1, u2) T if and only if there is no point xi = (xi1, xi2) T in the sample that is situated in the L-shaped risk region {x : x 6≤ u} (in gray) defined by u in its elbow. Points in the gray region are considered to exceed the multivariate threshold u. different story-line starting from blo… view at source ↗

**Figure 2.** Extremal regions denoted by {X 6≤ u} (left panel) and {X > u} (right panel) in a schematic bivariate example. tation at two distant locations at the same time is then relatively much smaller than the probability of such an event at one of the two locations. In the literature, this situation is referred to as asymptotic independence, and models developed in this chapter do not possess the correct lenses to measure t… view at source ↗

**Figure 3.** As the sample size n grows, the failure set B + log n escapes to +∞. As the sample size n grows, the number Nn(B) of sample points E1, . . . , En that fall in the risk region B +log n converges in distribution to a Poisson random variable with expectation Λ(B). probability of “success”. The translation by log n is chosen in such a way that the expected number of points in the failure set stabilizes: by Eq. (21), we … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The exponent measure ν on [0,∞) D is determined by the angular measure H on S = {x ≥ 0 : kxk = 1} by homogeneity via Equation (42). The set in gray is {x ≥ 0 : kxk > r, x/ kxk ∈ B}. In the picture, k · k is the Euclidean norm, but other norms can be used too. We think …

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Estimating the H\"usler--Reiss variogram matrix by clipped moments
math.ST 2026-07 conditional novelty 7.0 of 10

A lower-tail-clipped moment method estimates the Hüsler–Reiss variogram matrix with reduced bias under weak tail dependence while preserving asymptotic normality.

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages · cited by 1 Pith paper

[1]

Beirlant, Y

J. Beirlant, Y. Goegebeur, J. Teugels, and J. Segers. Statistics of Extremes . Wiley Series in Probability and Statistics. Chichester: John Wil ey & Sons Ltd., 2004, pp. xiv+490

work page 2004
[2]

Modelling extreme multivariate ev ents

S. Coles and J. Tawn. “Modelling extreme multivariate ev ents”. In: Journal of the Royal Statistical Society, Series B 53 (1991), pp. 377–392

work page 1991
[3]

Statistics of Extremes

A. Davison and R. Huser. “Statistics of Extremes”. In: Annual Review of Statistics and Its Application 2.1 (2015), pp. 203–235

work page 2015
[4]

Graphical models for extremes

S. Engelke and A. S. Hitz. “Graphical models for extremes ”. In: Journal of the Royal Statistical Society Series B: Statistical Methodolo gy 82.4 (2020), pp. 871– 932

work page 2020
[5]

M. Falk. Multivariate extreme value theory and D-norms . Springer Series in Oper- ations Research and Financial Engineering. Springer, Cham , 2019

work page 2019
[6]

M. Falk, J. Hüsler, and R.-D. Reiss. Laws of Small Numbers: Extremes and Rare Events. Third edition. Birkhäuser, 2011

work page 2011
[7]

A spectral representation for max-stable pr ocesses

L. de Haan. “A spectral representation for max-stable pr ocesses.” In: Ann. Probab. 12 (1984), pp. 1194–1204

work page 1984
[8]

de Haan and A

L. de Haan and A. Ferreira. Extreme Value Theory: An Introduction . New York: Springer-Verlag, 2006

work page 2006

Show all 19 references

[9]

Limit theory for multivariate sample extremes

L. de Haan and S. Resnick. “Limit theory for multivariate sample extremes”. In: Zeitschrift für Wahrscheinlichkeitstheorie und verwandt e Gebiete 40 (1977), pp. 317–337

1977
[10]

Statistical I nference for Hüsler–Reiss Graphical Models Through Matrix Completions

M. Hentschel, S. Engelke, and J. Segers. “Statistical I nference for Hüsler–Reiss Graphical Models Through Matrix Completions”. In: Journal of the American Sta- tistical Association https://doi.org/10.1080/01621459.2024.2371978 (2024)

2024
[11]

X-vine models for m ultivariate extremes

A. Kiriliouk, J. Lee, and J. Segers. “X-vine models for m ultivariate extremes”. In: Journal of the Royal Statistical Society Series B: Statisti cal Methodology https://doi.org/10.1093/jrsssb/qkae105 (2024), qkae10 5

2024 doi
[12]

Peaks over thresholds modeling with multivariate generalized Pareto distributi ons

A. Kiriliouk, H. Rootzén, J. Segers, and J. L. Wadsworth . “Peaks over thresholds modeling with multivariate generalized Pareto distributi ons”. In: Technometrics 61.1 (2019), pp. 123–135

2019
[13]

Max-linear models in ran dom environment

C. Klüppelberg and E. Sönmez. “Max-linear models in ran dom environment”. In: Journal of Multivariate Analysis 190 (2022), p. 104999

2022
[14]

Multivariat e generalized Pareto distri- butions along extreme directions

A. Mourahib, A. Kiriliouk, and J. Segers. “Multivariat e generalized Pareto distri- butions along extreme directions”. In: Extremes https://doi.org/10.1007/s10687- 024-00501-4 (2024). 32

2024 doi
[15]

S. Resnick. Extreme Values, Regular Variation and Point Processes . New York: Springer-Verlag, 1987

1987
[16]

S. Resnick. Heavy Tail Phenomena: Probabilistic and Statistical Model ing. New York: Springer, 2007

2007
[17]

Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws

S. Resnick. “Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws”. In: Stochastics: An Interna- tional Journal of Probability and Stochastic Processes 80 (2008), pp. 269–298

2008
[18]

Multivaria te peaks over thresholds models

H. Rootzén, J. Segers, and J. L. Wadsworth. “Multivaria te peaks over thresholds models”. In: Extremes 21.1 (2018), pp. 115–145

2018
[19]

Multivariate generalized P areto distributions

H. Rootzén and N. Tajvidi. “Multivariate generalized P areto distributions”. In: Bernoulli 12.5 (2006), pp. 917–930. 33

2006

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Beirlant, Y

J. Beirlant, Y. Goegebeur, J. Teugels, and J. Segers. Statistics of Extremes . Wiley Series in Probability and Statistics. Chichester: John Wil ey & Sons Ltd., 2004, pp. xiv+490

work page 2004

[2] [2]

Modelling extreme multivariate ev ents

S. Coles and J. Tawn. “Modelling extreme multivariate ev ents”. In: Journal of the Royal Statistical Society, Series B 53 (1991), pp. 377–392

work page 1991

[3] [3]

Statistics of Extremes

A. Davison and R. Huser. “Statistics of Extremes”. In: Annual Review of Statistics and Its Application 2.1 (2015), pp. 203–235

work page 2015

[4] [4]

Graphical models for extremes

S. Engelke and A. S. Hitz. “Graphical models for extremes ”. In: Journal of the Royal Statistical Society Series B: Statistical Methodolo gy 82.4 (2020), pp. 871– 932

work page 2020

[5] [5]

M. Falk. Multivariate extreme value theory and D-norms . Springer Series in Oper- ations Research and Financial Engineering. Springer, Cham , 2019

work page 2019

[6] [6]

M. Falk, J. Hüsler, and R.-D. Reiss. Laws of Small Numbers: Extremes and Rare Events. Third edition. Birkhäuser, 2011

work page 2011

[7] [7]

A spectral representation for max-stable pr ocesses

L. de Haan. “A spectral representation for max-stable pr ocesses.” In: Ann. Probab. 12 (1984), pp. 1194–1204

work page 1984

[8] [8]

de Haan and A

L. de Haan and A. Ferreira. Extreme Value Theory: An Introduction . New York: Springer-Verlag, 2006

work page 2006