An Offspring of Multivariate Extreme Value Theory: D-Norms. Michael Falk University of Wuerzburg, Germany London, December 2015

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1 205 CRoNoS Winter Course: An Offspring of Multivariate Extreme Value Theory: D-Norms Michael Falk University of Wuerzburg, Germany London, December 205 This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this

2 We do not want to calculate, we want to reveal structures. - David Hilbert, ii

3 Preface Multivariate extreme value theory MEVT is the proper toolbox for analyzing several extremal events simultaneously. Its practical relevance in particular for risk assessment is, consequently, obvious. But on the other hand MEVT is by no means easy to access; its key results are formulated in a measure theoretic setup, a fils rouge is not visible. Writing the angular measure in MEVT in terms of a random vector, however, provides the missing fils rouge: Every result in MEVT, every relevant probability distribution, be it a max-stable one or a generalized Pareto distribution, every relevant copula, every tail dependence coefficient etc. can be formulated using a particular kind of norm on multivariate Euclidean space, called D-norm. Norms are introduced in each course on mathematics as soon as the multivariate Euclidean space is introduced. The definition of an arbitrary D-norm requires only the additional knowledge on random variables and their expectations. But D-norms do not only constitute the fils rouge through MEVT, they are of mathematical interest of their own. In Sessions to 3 we provide in the introductory chapter the theory of D-norms in i

4 detail. The second chapter introduces multivariate generalized Pareto distributions and max-stable distributions via D-norms. The third chapter provides the extension of D-norms to functional spaces and, thus, deals with generalized Pareto processes and max-stable processes. Session 4, in addition to a brief summary of univariate EVT and D-norms, provides a relaxed tour through the essentials of MEVT, due to the D-norms approach. Quite recent results on multivariate records complete this text. ii

5 Contents Introduction. Norms and D-Norms Examples of D-Norms Takahashi s Characterizations Max-Characteristic Function Convexity of the Set of D-Norms D-Norms and Copulas Normed Generators Theorem Metrization of the Space of D-Norms Multivariate Generalized Pareto and Max Stable Distributions Multivariate Simple Generalized Pareto Distributions Multivariate Max-Stable Distributions Standard Multivariate Generalized Pareto Distribution Max-Stable Random Vectors as Generators of D-Norms iii

6 3 The Functional D-Norm Introduction Generalized Pareto Processes Max-Stable Processes Tutorial: D-Norms & Multivariate Extremes Univariate Extreme Value Theory Multivariate Extreme Value Distributions Extreme Value Copulas et al A Measure of Tail Dependence Takahashi s Theorem Some General Remarks on D-Norms Functional D-Norm Some Quite Recent Results on Multivariate Records iv

7 Chapter Introduction. Norms and D-Norms General Definition of a Norm Definition... A function f : Rd [0, is a norm, if it satisfies for all x, y Rd, λ R f x 0 x 0 Rd, f λx λ f x, f x + y f x + f y Condition.2 is called homogeneity and condition.3 is called triangle inequality or -inequality, for short.

8 A norm f : R d [0, is typically denoted by x fx, x R d..4 Each norm on R d defines a distance, or metric on R d via dx, y x y, x, y R d..5 Well known examples of norms are the sup-norm and the L -norm x : max i d x i.6 x : The Logistic Norm d x i, x x,..., x d R d..7 i Not that obvious is the logistic family d /p x p : x i p, p <..8 The corresponding -inequality i is known as the Minkowski-inequality. cf. Rudin 976, Proposition 3.5 fx + y fx + fy 2

9 Lemma... We have for p q and x Rd i kxkp kxkq, ii lim kxkp kxk. p Proof. i The inequality is obvious for q : kxk P d q i xi /q. Now consider p q < and choose x 6 0 Rd. Put S : kxkp. Then we have x S p and we have to establish x. S q As xi [0, ] S and thus xi S q xi S p, 3 i d,

10 we obtain x d q q d p q xi xi S q S S i which is i. ii We have, moreover, for x 0 R d and p [, i x p q S p p q, x x p which implies ii. d p p xi i x x d p x p x, Norms by Quadratic Forms Let A a ij i,j d be a positive definite d d-matrix, i.e., the matrix A is symmetric, A A a ji i,j d, and satisfies x Ax x i a ij x j > 0, x R d, x 0 R d. Then defines a norm on R d. i,j d x A : x T Ax 2, x R d, 4

11 0 With A we obtain, for example, x 0 A x 2 + x 2 2 /2 x 2. Conditions. and.2 are obviously satisfied. The -inequality follows by means of the Cauchy-Schwarz inequality 2 as follows: x Ay 2 x Ax y Ay, x, y R d, x + y 2 A x + yt A x + y Definition of D-Norms x T Ax + y T Ax + x T Ay + y T Ay x T Ax + 2x T Ax 2 y T Ay 2 + y T Ay 2 x T Ax 2 + y T Ay 2. Let now Z Z,..., Z d be a random vector rv, whose components satisfy Then Z i 0, EZ i, i d. x D : E max x i Z i, x R d, i d defines a norm, called D-norm and Z is called generator of x D. 2 cf. Rudin 976 5

12 i.e., The homogeneity condition.2 is obviously satisfied. Further, we have the bounds x max i d x i max E x i Z i i d E max x i Z i i d d E i x i Z i d x i EZ i i x, x R d, x x D x, x R d..9 This implies condition.. The inequality is easily seen by x + y D E max x i + y i Z i i d E max x i + y i Z i i d E max x i Z i + max y i Z i i d i d 6

13 E max x i Z i + E max y i Z i i d i d x D + y D. Basic Properties of D-Norms Denote by e j 0,..., 0,, 0,..., 0 R d the j-th unit vector in R d, j d. Each D-norm satisfies e j D E max δ ijz i EZ j, i d where δ ij if i j and zero elsewhere, i.e., each D-norm is standardized. Each D-norm is monotone, i.e., we have for 0 x y, where this inequality is taken componentwise, x D E max x iz i E max y iz i y D. i d i d There are norms that are not monotone: Choose for example δ A δ with δ, 0. The matrix A is positive definite, the norm x A x T Ax 2 x 2 + 2δx x 2 + x is not monotone; just compare x, x 2 with x, x 2 + ε. Each D-norm is obviously radial symmetric, i.e., changing the sign of arbitrary components of x R d does not alter the value of x D. This means that the values of a 7

14 D-norm are completely determined by its values on the subset {x R d : x 0}. The above norm A does not have this property..2 Examples of D-Norms The two Extremal D-Norms Choose the constant generator Z :,,...,. Then x D E max x i Z i i d E max x i x, i d i.e., the sup-norm is a D-norm. Let X 0 be a rv with EX and put Z : X, X,..., X. Then Z is a generator of the D-norm x D Emax i d x i Z i Emax i d x i X max i d x i EX x EX x. 8

15 This example shows that the generator of a D-norm is in general not uniquely determined, even its distribution is not. Let now Z be a random permutation of d, 0,..., 0 R d with equal probability /d, i.e., { d, with probability /d Z i 0, with probability /d, i d, and Z + + Z d d. The rv Z is consequently the generator of a D-norm: x D E max x i Z i i d d E max x i Z i {Zj d} i d j d E max x i Z i {Zj d} i d j d E x j d {Zj d} d j j x j d E {Zj d} 9

16 d X xj d P Zj d j d X xj j kxk, i.e., k k is a D-norm as well. Inequality.9 shows that the sup-norm k k is the smallest D-norm and that the L -norm k k is the largest D-norm. Each Logistic Norm is a D-Norm Pd p /p Proposition.2.. Each logistic norm kxkp x, p <, is a D-norm. i i For < p < a generator is given by X Xd Z Z,..., Zd,...,, Γ p Γ p where X,..., Xd are independent and identically iid Fr echet-distributed rv, i.e., P Xi x exp x p, with EXi Γ p, i d. 0 x > 0, i,..., d,

17 Γs 0 t s exp t dt, s > 0, denotes the Gamma function. Proof. Put for notational convenience µ : EX Γ p. From the fact that the expectation of a non-negative rv X is in general given by 0 P X > t dt use Fubini s theorem, we obtain E max x i Z i P max x i Z i > t dt i d 0 i d P max x i Z i t dt 0 i d P Z i t 0 x i, i d dt d P Z i t dt 0 x i i d p xi exp dt 0 tµ i d i exp x i p dt tµ p 0 d The substitution t t i x i p p /µ now implies that the integral above equals d i x i p p µ 0 exp t p dt x p P X > t dt EX 0

18 kxkp EX EX kxkp..3 Takahashi s Characterizations Takahashi s Characterizations of k k and k k Theorem.3. Takahashi 988. Let k kd be an arbitrary D-norm on Rd. Then we have the equivalences k kd k k y > 0 Rd : kykd kyk, k kd k k kkd Corollary.3.. We have for an arbitrary D-norm k kd on Rd k k k kd kkd. k k d Proof. To prove Theorem.3. we only have to show the implication. Let Z,..., Zd be a generator of k kd. 2

19 i Suppose we have y D y for some y > 0 R d, i.e., This leads to d E y i Z i E i d i d i Emax i d y iz i d i y i d y i EZ i i d E y i Z i. i d max y iz i E i d y i Z i max i d y iz i 0 y i Z i max i d y iz i a.s. i y i Z i max i d y iz i } {{ } 0 a.s. almost surely Hence Z i > 0 for some i {,..., d} implies Z j 0 for all j i, and we have for arbitrary 3 0

20 x 0 d x i Z i max x iz i i d i d E x i Z i E i x x D. ii We have the following list of conclusions: a.s. max x iz i i d,..., D E max Z i EZ j, j d, i d E max Z i Z j 0, j d, i d }{{} 0 max Z i Z j 0 a.s., j d, i d Z Z 2... Z d max Z i i d E max x i Z i E max x i Z i d i d E x Z x EZ 4

21 kxk, x Rd. Theorem.3. can easily be generalized to sequences of D-norms. Theorem.3.2. Let k kdn, n N, be a sequence of D-norms on Rd. i x Rd : kxkdn kxk y > 0 : kykdn kyk n n ii x Rd : kxkdn kxk kkdn n n Corollary.3. carries over. n n Proof. Let Z,..., Zd be a generator of k kdn. Again we only need to show the implication. d i n We suppose kyk kyk o Dn n 0 for some y > 0 R. With the notation Mj : n yj Z j n max i d yi Zi we get for every j,..., d kyk kykdn d X n n y Z max yi Zi E i d i i i {z } 0!! d X n n yi Zi max yi Zi M j E i d i 5

22 d E i i j d y i E i i j y i Z n i Mj Z n i Mj n 0 as the left hand side of this equation converges to zero by assumption: y y D n n 0. Since y i > 0 for all i,..., d, we have E Z n i Mj n 0.0 for all i j. Now take an arbitrary x R d. From.9 we know that known that 0 x x D n d E x i Z n i max x i Z n i i d i }{{} 0 d d E x i Z n i max x i Z n i Mj i d j i d d E j i x i Z n i 6 max x i Z n i i d Mj

23 d X d X j i i6j xi E n Z i M j n 0 {z } n 0 by.0 x Rd : kxkdn n kxk. ii We use inequality.9 and obtain 0 kxkdn kxk n E max xi Zi max xi i d i d n max xi E max Zi max xi i d i d i d n kxk E max Zi i d kxk kkdn 0. n Theorem.3.3. Let k kdn, n N, be a sequence of D-norms on Rd. We have i k kdn k k i < j d : kei + ej kdn 2 n n 7

24 ii k kdn k k i {,..., d} j 6 i : kei + ej kdn. n n Proof. i For all i < j d we have 2 kei + ej kdn n n n n E Zi + Zj E maxzi, Zj n n n n E Zi + Zj maxzi, Zj n n n n E Zi + Zj maxzi, Zj {Z n max k d Z n } j k n E Zi {Z n max k d Z n } 0. j k n Therefore E Zi {Z n max k d Z n } 0, which is.0 for y. We can repeat the j k n remaining steps of the preceding proof and get the desired assertion. ii For our given value of i we have 0 kkdn n n E max Zk Zi k d d X E n max Zk Zi E max n n Zi, Zj j 8 {Z n max k d Z n } k d j d X n j n Zi k {Z n max k d Z n } j k

25 d X E max n n Zi, Zj n Zi j d X kei + ej kdn n 0 j j6i which proves the assertion by part ii of Theorem.3.2. Corollary.3.2. Let k kd be an arbitrary D-norm on Rd. i k kd k k i < j d : kei + ej kd 2 kei + ej k ii k kd k k i {,..., d} j 6 i : kei + ej kd kei + ej k Proof. Put k kdn k kd in the preceding theorem. Remark.3.. Choose i < j d. Note that kx, ykdi,j : kxei + yej kd, x, y R, defines a D-norm on R2 with generator Zi, Zj, where Z,...Zd generates k kd. From Takahashi s Theorem, part i, we obtain that the condition i < j d : kei + ej kd 2 9

26 is equivalent with the condition x, y R, i < j d : kxei + yej kd kxei + yej k x + y..4 Max-Characteristic Function The Max-Characteristic Function of a Generator Recall that the generator of a D-norm is not uniquely determined, even its distribution is not. Lemma.4. Balkema. Let X X, X2,..., Xd 0, Y Y, Y2,..., Yd 0 be rv with EXi, EYi <, i d. If we have for each x > 0 Rd E max, x X,..., xd Xd E max, x Y,..., xd Yd, then X D Y, where D denotes equality in distribution. Proof. We have for x > 0 and c > 0 Xd X,..., E max, cx cx d Z X Xd P max,,..., t dt cx cxd 0 Z P t, Xi tcxi, i d dt 0 20

27 Z P Xi tcxi, i d dt + The substitution t 7 t/c yields that the right-hand side above equals Z + P Xi txi, i d dt. c c Repeating the preceding arguments with Yi in place of Xi, we obtain from the assumption that for all c > 0 Z P Xi txi, i d dt c Z P Yi txi, i d dt. c Taking right derivatives with respect to c we obtain for c > 0 P Xi cxi, i d P Yi cxi, i d, and, thus, the assertion. Corollary.4.. If, Z,..., Zd is the generator of a D-norm, then the distribution of Z,..., Zd is uniquely determined. Take, for example, Z,..., Rd, which generates the sup-norm k k on Rd. Then, Z generates the sup-norm on Rd+. 2

28 Let, on the other hand, Z be a random permutation of d, 0,..., 0 R d, which generates. The D-norm generated by, Z on R d+ is x D E max x, x 2 Z 2,..., x d+ Z D d E max x, x 2 Z 2,..., x d+ Z D Z i d d i d E max x, x 2 Z 2,..., x d+ Z D Z i d i d max x, d x i i2 d max i2 x d, x i. Let Z Z,..., Z d be the generator of a D-norm D. Then we call ϕx : Emax, x Z,... x d Z d, x R d, the max-characteristic function of Z. The max-characteristic function of the random permutation of d, 0,..., 0, for instance, is d ϕx max d, x i, x R d. i 22

29 The max-characteristic function of,..., is ϕx max, kxk, x Rd. We have in general ϕx Emax, x Z,..., xd Zd E max xi Zi kxkd i d and, thus, 0 ϕx kxkd Emax, x Z,..., xd Zd max xi Zi i d E max xi Zi {max i d xi Zi <}, i d x Rd, if Z generates the D-norm k kd..5 Convexity of the Set of D-Norms The Set of D-Norms is Convex Proposition.5.. The set of D-norms on Rd is convex, i.e., if k kd and k kd2 are D-norms, 23

30 then k kλd + λd2 : λ k kd + λ k kd2 is for each λ [0, ] a D-norm as well. Take, for example, the convex combination of the two D-norms k k and k k : λ kxk + λ kxk λ max xi + λ i d d X xi. i This is the Marshall-Olkin D-norm with parameter λ [0, ]. Proof of Proposition.5.. Let ξ be a rv with P ξ λ P ξ 2 and independent of Z and Z 2, where Z, Z 2 are generators of k kd, k kd2. Then Z : Z ξ is a generator of k kλd + λd2, as we have for x 0! 2 X ξ ξ E max xi Zi E max xi Zi {ξj} i d i d j 2 X E E i d j 2 X j max j x i Zi i d j 2 X max ξ x i Zi E max i d j xi Zi 24 {ξj} {ξj} E {ξj}

31 λe max x iz i + λe max x iz 2 i. i d i d A Bayesian Type of D-Norms The preceding convexity of the set of D-norms can be viewed as a special case of a Bayesian type D-norm as illustrated { by the} following example. Consider the logistic family p : p of D-norms as defined in.8. Let f be a probability density on [,, i.e., f 0 and fp dp. Then x f : x p fp dp, x R d, defines a D-norm on R d. This can easily be seen as follows. Let X be a rv on [, with this probability density f and suppose that X is independent from each generator Z p of p, p. Then Z f : Z X generates the D-norm f : and EZ f E EZ X X pfp dp max x i Z f,i i d 25 EZ p fp dp

32 E max x i Z X,i i d E max x i Z X,i X p i d x p fp dp. fp dp If we take, for instance, the Pareto density f λ p : λp +λ, p, with parameter λ > 0, then we obtain p /p x fλ x i p λp +λ dp, x R d. i The convex combination of two arbitrary D-norms can obviously be embedded in this Bayesian type approach..6 D-Norms and Copulas D-Norms and Copulas Let the rv U U,..., U d follow a copula, i.e., each component U i is uniformly distributed on 0,. As EU i 0 u du /2, the rv Z : 2U is generator of a D-norm. But not every D-norm can be generated this way: take, for example, d 2 and x, y x + y. Suppose that there exists a rv U U, U 2 following a copula such 26

33 that x, y 2E max x U, y U 2, x, y R. Putting x y we obtain and, thus, But 2 2E maxu, U 2 }{{} [0,] P maxu, U 2. P maxu, U 2 P {U } {U 2 } P U + P U 2 0. It is, moreover, obvious, that on R d with d 3 cannot be generated by 2U, as,..., d > 2E max i d U i. There are consequently strictly more D-norms than copulas..7 Normed Generators Theorem By T we denote in what follows the number of elements in a set T. The following auxiliary result can easily be proved by induction, just use the equation minmaxa,..., a n, a n+ maxmina, a n+,..., mina n, a n+. 27

34 Lemma.7.. We have for arbitrary numbers a,..., an R : X T min ai, maxa,..., an i T 6T {,...,n} X mina,..., an T max ai. i T 6T {,...,n} Corollary.7.. If Z, Z 2 generate the same D-norm, then 2 E min xi Zi E min xi Zi, i d i d Proof. Corollary.7. can be seen as follows: E min xi Zi E i d X X 6T {,...,d} T max xi Zj j T 6T {,...,d} x Rd. T E max xi Zj j T X X T xj ej j T 6T {,...,d} D 28

35 T {,...,d} E E T {,...,d} min i d T E max j T j T x i Z 2 j T max x i Z 2 j x i Z 2 i. Dual D-Norm Function Let D be an arbitrary D-norm on R d with arbitrary generator Z Z,..., Z d. Put x D : E min x i Z i, x R d, i T which we call the dual D-norm function corresponding to D. It is independent of the particular generator Z, but the mapping D D is not one-to-one. In particular we have that D 0 is the least dual D-norm function, corresponding to D, and x D min i d x i x, x R d, 29

36 is the largest dual D-norm function, corresponding to k kd k k, i.e., we have for an arbitrary dual D-norm function the bounds 0 oo oo oo ood oo oo. While the first inequality is obvious, the second one follows from xk E xk Zk E min xi Zi, k d. i d The Exponent Measure Theorem The following result is based on the characterization of a max-infinite divisible df in Balkema and Resnick 977. We, therefore, call it Exponent Measure Theorem. Put E : [0, \ {0} Rd and tb : {tb : b B} for an arbitrary set B E and t > 0. Theorem.7. Exponent Measure Theorem. Let k kd be an arbitrary D-norm on Rd. Then ν [0, x]{ E : x 0, x 6 0, x, D with the convention k/xkd, if some component of x is zero, defines a measure ν on 30

37 E, which satisfies for each Borel subset B of E ν tb ν B, t t > 0. Sketch of the proof. Let Z,..., Zd be a generator of the D-norm k kd and put for x E and 6 T {,..., d} νπi > xi, i T : E min Zi, i T xi with the convention 0/0, where πi y yi denotes the projection of y E onto its i-th component. Note that by Corollary.7. the value of E mini T Zi /xi does not depend on the special choice of the generator of k kd. The function ν is defined on a family of subsets of E, which is -stable and which generates the Borel σ-field BE in E. In order to extend it to a uniquely determined measure ν on BE, it has to satisfy νa, b] 0 for a, b E, a b. This will be shown below. From the well known inclusion exclusion principle we obtain for 0 < a b the equation! d [ νa, b] ν a, \ {πi > bi } ν a, \ i d [! {πi > bi ; πj > aj, j 6 i} i ν a, ν d [! {πi > bi ; πj > aj, j 6 i} i ν a, 3

38 E min i d T {,...,d} T {,...,d} ai Z i T {,...,d} T E mi E m {0,} d E m {0,} d m {0,} d T νπ i > b i, i T ; π j > a j, j T T E min mi min i d min i d min i T Zi min i d b i Zi b i min min i T Z i, min b i j T mi Zi a i Z j a j mi Z i, min b i j T mi mi Zi. a i Z j a j We claim that the integrand in the above expectation is nonnegative, i.e., we claim that for R d 0 x y mi x m i i y m i i 0.. Let U be a rv which follows the uniform distribution on 0,, and put U U,..., U R d. The df of U is F U u : P U u min i d u i, u [0, ] d. 32

39 We, thus, obtain for 0 u v R d by the well known inclusion exclusion principle P U u, v] mi F U u m v m,..., u m d m {0,} d 0. m {0,} d mi min i d u m i i v m i i d v m d d This implies inequality. by a proper scaling of x, y. We have, moreover, ν t a, b] t ν a, b], t > 0. The equality ν B : νtb t νb : ν 2 B, thus, holds on a generating class closed under intersections and is, therefore, true for any Borel subset B of E 3. Finally, we have for x > 0 R d by the inclusion exclusion principle and Lemma.7. ν 3 cf. Bauer 200, p.32 Remarks [0, x] E ν d {π i > x i } i T {,...,d} T {,...,d} E T {,...,d} T νπ i > x i, i T T E T min min Z i i T x i i T Z i x i 33

40 E max Zi i d xi x. D Existence of Normed Generators The proof of the following theorem is essentially the proof of the de Haan-Resnick representation4 of a multivariate max-stable df with unit Fr echet margins. Theorem.7.2 Normed Generators. Let k k be an arbitrary norm on Rd. For any D-norm k kd on Rd there exists a generator Z with the additional property kzk const. The distribution of this generator is uniquely determined. d 2 Corollary.7.2. For any D-norm on R there exist generators Z, Z with the property Pd 2 i Zi d and max i d Zi const. Proof. Choose k k k k in Theorem.7.2. Then d X Zi. const Z 4 de Haan and Resnick i

41 Taking expectations on both sides yields const Choose for the second assertion. d E i Z i d. Proof of Theorem.7.2. Let D be an arbitrary norm on R d. From the Exponent Measure Theorem.7. we know that ν [0, x] E : x, x 0 R d, x 0, D defines a measure ν on E with the property νtb t νb for each Borel subset B of E [0, x \{0} and each t > 0. Denote by S E : {z E : z } the unit sphere in E with respect to the norm. From the equality νtb t νb we obtain for t > 0 and any Borel subset A of S E { ν x E : x t, { ν ty E : y, { ν t y E : y, t ν {y E : y, 35 x x A } } y y A } y y A y y A }

42 : ΦA.2 t where Φ is the angular measure on S E corresponding to. Define the one-to-one function T : E [0, S E by x T x x,, x which is the transformation of a vector x on to its polar coordinates with respect to the norm. From.2 we obtain that the measure ν T B : νt B, induced by ν and T, satisfies ν T t, A ν {x E : T x t, A} { } x ν x E : x > t, x A and, hence, t ΦA A t, t, A ν T B dr dφ r2 dr dφ r2 B r 2 dr dφ..3 36

43 We have ν [0, x] ν T T [0, x] ν T T [0, x] with T [0, x] T {y E : y i > x i for some i d} {r, a 0, S E : ra i > x i for some i d} { } xi r, a 0, S E : r > min i d with the temporary convention 0/0. Hence, we obtain from equation.3 x ν [0, x] D ν T T [0, x] { } x i r, a 0, S E : r > min i d a i ν T { r,a 0, S E : r> min i d S E s 2 ds Φda x min i i d a i x S E min i i d a i Φda 37 } s 2 ds dφ x i a i a i

44 a i max Φda S i d E x i now with the convention 0/0 0 in the bottom line. Note that Φ is a finite measure as can be seen as follows. Choose in the preceding equation x i and let x j for j i. Then, by the fact that e i D, i d, we obtain a i Φda, i d..4 S E The finiteness of Φ now follows from the fact, that all norms on R d are equivalent: d d a i Φda i.e., ΦS E <. Put Then S E i a Φda S E const a Φda S E }{{} const ΦS E, m : ΦS E 0, Q : Φ m 38

45 defines a probability measure on S E. Let the rv X X,... X d S E follow this probability measure, i.e. P X Q. Then we have for Z : mx as well as and by.4. Finally, we have E max i d Z mx m X m Z 0, EZ i EmX i mex i m a i P X dx S E m a i Qda S E Φda m a i S E m m a i Φda m S E. Z i E m max x i i d 39 X i x i a.s.

46 Z ai P X,..., Xd da i d xi SE Z ai m max Qda i d xi SE Z ai Φda m max i d xi m Z SE ai max Φda i d xi SE x. D m max Example.7.. Put Z :,..., and Z 2 : X,..., X, where X 0 is a rv with EX. Both generate the D-norm k k, but only Z satisfies Z d. Example.7.2. Let V,..., Vd be independent and identically gamma distributed rv with Rd with components density γα x : xα exp x/γα, x > 0, α > 0. Then the rv Z Z i : Vi, V + + Vd i,..., d, follows a symmetric Dirichlet distribution Dirα on the closed simplex S d {u 0 Rd : 40

47 Pd i ui }, see Ng et al. 20, Theorem 2.. We obviously have EZ i /d and, thus, Z : dz.5 is a generator of a D-norm k kdα on Rd, which we call the Dirichlet D-norm with parameter α. We have in particular kzk d. d Pd P It is well-known that for a general α > 0 the rv Vi / j Vj and the sum dj Vj are i independent, see, e.g., the proof of Theorem 2. in Ng et al. 20. As EV + +Vd dα, we obtain for x x,..., xd Rd kxkdα E max xi Zi i d max i d xi Vi de V + + Vd max i d xi Vi EV + + Vd E α V + + Vd E max xi Vi. i d α A generator of k kdα is, therefore, also given by α V,..., Vd. 4

48 .8 Metrization of the Space of D-Norms Metrization of the Space of D-Norms Denote by Zk kd the set of all generators of a given D-norm k kd on Rd. Theorem.7.2 implies the following result. Lemma.8.. Each set Zk kd contains a generator Z with the additional property kzk d. The distribution of this Z is uniquely determined. Let P be the set of all probability measures on Sd : {x 0 Rd : kxk d}. By the preceding lemma we can identify the set D of D-norms on Rd with the subset PD of those R probability distributions P P which satisfy the additional condition Sd xi P dx, i,..., d. Denote by dw P, Q the Wasserstein metric between two probability distributions on Sd, i.e., dw P, Q : inf {E kx Y k : X has distribution P, Y has distribution Q}. As Sd, equipped with an arbitrary norm k k, is a complete separable space, the metric space P, dw is complete and separable as well; see, e.g., Bolley

49 Lemma.8.2. The subspace PD, dw of P, dw is also separable and complete. Proof. Let Pn, n N, be a sequence in PD, which converges with respect to dw to P P. We show that P PD. Let the rv X have distribution P and let X n have distribution Pn, n N. Then we have Z d Z d Z X X x P dx x P dx x P dx i i i n i Sd Sd i d X Sd n E Xi X i i E d X n Xi Xi! i n E X X, n N. As a consequence we obtain d Z X dw P, Pn n 0, x P dx i i Sd and, thus, P PD. The separability of PD can be seen as follows. Let P be a countable and dense subset of P. Identify each distribution P in P with a rv Y on Sd that follows this distribution P. Put Z Y /EY, where we can assume that each component of Y has positive expectation. This yields a countable subset of PD, which is dense. 43

50 We can now define the distance between two D-norms k kd, k kd2 on Rd by dw k kd, k kd2 n 2 : inf E Z Z : o i i Z generates k kdi, Z d, i, 2. The space D of D-norms on Rd, equipped with the distance dw, is by Lemma.8.2 a complete and separable metric space. Convergence of D-Norms and Weak Convergence of Generators For the rest of this section we restrict ourselves to generators Z of D-norms on Rd that satisfy kzk d. Proposition.8.. Let k kdn, n N {0}, be a sequence of D-norms on Rd with corresponding generators Z n, n N {0}. Then we have the equivalence dw k kdn, k kd0 n 0 Z n D Z 0, where D denotes ordinary convergence in distribution. Proof. Convergence of probability measures Pn to P0 with respect to the Wasserstein-metric is 44

51 equivalent with weak convergence together with convergence of the moments Z Z kxk Pn dx n kxk P0 dx, Sd Sd see, e.g., Villani But as we have for each probability measure P PD Z Z kxk P dx d P dx d, Sd Sd convergence of the moments is automatically satisfied. Lemma.8.3. We have for arbitrary D-norms k kd, k kd2 on Rd the bound kxkd kxkd2 + kxk dw k kd, k kd2 and, thus, sup x Rd,kxk r kxk r dw k k, k k kxk D D2 D D2, Proof. Let Z i be a generator of k kdi, i, 2. We have kxkd E max xi Zi i d 2 2 E max xi Zi + Zi Zi i d 45 r 0.

52 which implies the assertion. E max i d x i Z 2 i x D2 + x E + x E max Z Z 2, Z i d i Z 2 i 46

53 Chapter 2 Multivariate Generalized Pareto and Max Stable Distributions 2. Multivariate Simple Generalized Pareto Distributions Multivariate Simple Generalized Pareto Distributions Let Z Z,..., Z d be a generator of a D-norm D with the additional property Z i c, i d, 2. for some constant c, see Corollary.7.2. Let U be a rv that is uniformly distributed on 0, and which is independet of Z. Put V V,..., V d : U Z,..., Z d : U Z. 47

54 Note that for x > P U x P x U x, i.e., /U follows a standard Pareto distribution with parameter. We have, moreover, for x > c and i d by Fubini s theorem P U Z Zi i x P x U Zi E x U z [0,] [0,c] x u P U, Z i du, z z x u P U P Z i du, z [0,] [0,c] c 0 c 0 c 0 P 0 x z x u P Udu P Z i dz z x U P Z i dz z x P Z i dz c 0 x E Z i z P Z i dz 48

55 x, 2.2 where P X denotes the distribution of a rv X, and P X, Y P X P Y if the rv X, Y are independent. The product Z i /U, therefore, follows in its upper tail a standard Pareto distribution. The special case Z i yields the standard Pareto distribution everywhere. We call the distribution of V Z/U a d-variate simple generalized Pareto distribution simple GPD. The Distribution Function of a GPD By repeating the arguments in equation 2.2 we obtain for x c,..., c c Zi P V x P U x i, i d Zi P U, i d x i P U z i, i d P Zdz,..., z d [0,c] x d i z i P U max P Zdz,..., z d [0,c] d i d x i z i max P Zdz,..., z d [0,c] d i d x i

56 z i max [0,c] d i d E max i d x, D Z i x i P Zdz,..., z d x i i.e., the multivariate distribution function df of V /x D. is in its upper tail given by The Survival Function of a GPD By repeating the arguments in the derivation of equation 2.3 again, we obtain for x c P V x P U z i [0,c] d P [0,c] d P [0,c] d, i d x i U z i, i d P Zdz,..., z d x i z i U min P Zdz,..., z d i d x i z i min P Zdz,..., z d i d x i 50

57 Z i E min i d x i /x D. 2.4 An Application to Risk Assessment Suppose that the joint random losses of a portfolio consisting of d assets are modelled by the rv V. The probability that the d losses jointly exceed the vector x > c is by equation 2.4 given by Z i P V x E min. i d x i If we suppose that D, then we can choose the constant function Z,... as a generator and, thus, P V x min i d x i max x, x,...,. i i d If we suppose that D, then we can choose the random permutation of d, 0,..., 0 with equal probability /d as a generator Z. in this case we have min Z i 0 i d and, thus, Z i P V x E min 0, x d,..., d. i d x i 5

58 This example shows that assessing the risk of a portfolio is highly sensitive to the choice of the stochastic model: For x d,..., d and D, the probability for the losses jointly exceeding the value d is /d, whereas for D it is zero! Risk assessment has, consequently, become a major application of extreme value analysis in recent years. 2.2 Multivariate Max-Stable Distributions Introducing Multivariate Max-Stable Distributions Let now V V,..., V d, V 2 V 2,..., V 2 d rv V Z/U. Then we obtain for the vector of the componentwise maxima max V i : i n max V i, max V i 2,..., max V i i n i n i n d from equation 2.3 for x > 0 and n large such that nx > c P n max V i x i n P max V i nx i n P V i nx, i n n P V i nx i 52,... be independent copies of the 2.5

59 P V nx n n nx D n n x D n n x D n exp x : Gx, x > 0 R d, D where /x is meant componentwise, i.e., /x /x,..., /x d. Suppose that at least one component of x is equal to zero, say component i 0. Then P V nx P V i0 nx i0 Zi0 P U 0 P Z i0 0 P Z i0 0 < by the fact that EZ i0. As a consequence we obtain in this case P n max V i x P V nx n i n P Z i0 0 n n 0 53

60 We, thus, have where P n max V i x Gx, x i n n Rd, { exp x Gx D, if x > 0, 0 elsewhere. As P n max V i, n N, is a sequence of df on R d, it is easy to check that its i n limit G is a df itself. It is obvious that the df G satisfies G n nx exp n nx exp D x Gx, D i.e., G n nx Gx, x R d, n N, which is the so called max-stability of G: Let the rv ξ R d have df G and let ξ, ξ 2,... be independent copies of ξ. Then we have for the vector of componentwise maxima Falk et al. 20, Section 4. P n max i n ξi x P max i n ξi nx P ξ i nx, i n P ξ nx n 54

61 G n nx Gx, x R d, which explains the name max-stability. The Simple Multivariate Max-Stable Distribution By keeping x i > 0 fixed and letting x j tend to infinity for j i, we obtain the marginal distribution of G: G i x i :P ξ i x i lim x j j i lim x j j i P ξ i x i, ξ j x j, j i Gx lim exp x j x j i D lim exp x j,...,,..., D x x i x j i d exp 0,..., 0,, 0,..., 0 x i D exp E Z i x i 55

62 exp xi. Each univariate marginal df of G is, consequently, G F x : exp, x > 0, x which is the Fréchet df with parameter, or unit Fréchet df for short. We call the multivariate df G with unit Fréchet margins multivariate simple maxstable. The Standard Multivariate Max-Stable Distribution Let the rv ξ R d follow a multivariate simple max-stable df, i.e., P ξ x exp /x D, x > 0. Put η ξ ξ,..., ξ d and note that P ξ i 0 0, i d. Then we obtain for x 0 R d P η x P x i, i d ξ i P ξ i, i d x i P ξ x 56

63 By putting for x R d exp x D : G D x. G D x : exp minx, 0,..., minx d, 0 D, we obtain a df on R d, which is max-stable as well: x G D x, x R n d, n N. G n D G n D /n is the df of n max i n η i, where η, η 2,... are independent copies of η. Note that each univariate margin of G D is the standard negative exponential df: P η i x P η xe i exp xe i D exp x e i D expx, x 0. We call G D multivariate standard max-stable SMS. Takahashi revisited Let the rv η η,..., η d have in what follows the SMS df P η x Gx exp x D, x 0 R d, with an arbitrary D-norm D on R d. Theorem.3. can now be formulated as follows. 57

64 Theorem With η as above we have the equivalences i η,..., ηd are independent d y < 0 R : P ηi yi, i d d Y P ηi yi. i ii η η2 ηd a.s. P η, η2,..., ηd. Proof. The assumption η,..., ηd are independent is equivalent with the condition k kd k k. The assumption η η2 ηd a.s. is equivalent with the condition k kd k k. The assertion is, therefore, an immediate consequence of Theorem.3.. The following characterization is an immediate consequence of Theorem.3.3. Note that for arbitrary i < j d P ηi, ηj P ηi, ηj, ηk 0, k 6 {i, j} exp kei + ej kd. Part ii is, obviously, trivial. We list it for the sake of completeness. Theorem With η as above we have the equivalences i η,..., ηd are independent η,..., ηd are pairwise independent. 58

65 ii η η2 ηd a.s. η,..., ηd are pairwise completely dependent. The distribution of an arbitrary d-variate max-stable rv can be obtained by means of η as above together with a proper non random transformation of each component ηi, i d, see, e.g., Falk et al. 20, equation The preceding characterizations, therefore, carry over to an arbitrary multivariate max-stable rv see Standard Multivariate Generalized Pareto Distribution Standard Multivariate Generalized Pareto Distribution Choose K < 0 and put W : W,..., Wd U U : max, K,..., max, K, Z Zd where U is uniformly distributed on 0, and independent of the generator Z of the D-norm k kd, which is bounded by c. The additional constant K avoids division by zero. Repeating the arguments in equation 2.3 we obtain P W x kxkd, where x0 < 0 Rd depends on K and c. 59 x0 x 0 Rd,

66 Repeating the arguments in equation 2.5 one obtains P n max i n W i x n exp x D, x 0 Rd, where W, W 2,... are independent copies of W. We call a df H on R d a standard GPD, if there exists x 0 < 0 R d such that Hx x D, x 0 x 0 R d. Note that the i-th marginal df H i of H is given by H i x xe i D x e i D + x, x 0i x 0, i d, which coincides on [x 0i, 0] with the uniform df on [, 0]. 2.4 Max-Stable Random Vectors as Generators of D-Norms Max-Stable Random Vectors as Generators of D-Norms Let the rv η η,..., η d follow the SMS df Gx P η x exp x D, x 0 R d. Choose c 0,. Then the rv / η i c has the df P η i c x P x η i c P x η /c i 60

67 P exp x η /c i x /c, x > 0, i d, i.e. / η i c follows the Fréchet df F α x exp x α, x > 0, with parameter α /c; note that P η i 0 0. Its expectation is E η i c x exp x α x α α dx 2.6 α Γ x α exp x α dx x α exp xdx x α exp xdx α Γ c : µ c The rv Z Z,..., Z d : µ c η c,..., η d c 2.7 now satisfies Z i 0 and EZ i, i d, i.e., Z is the generator of a D-norm. Can we specify it? 6

68 Note that the rv / η c,..., / ηd c follows a max-stable df with Fr echet-margins: xi, i d Hx P ηi c! P ηi /c, i d x i!!, x > 0 Rd, exp /c,..., /c x xd D and for each n N:!n n c H n x exp,..., nc x /c nc xd /c D!! n exp /c,..., /c n x xd D Hx, x > 0 Rd. c c Now we can specify the D-norm, which is generated by Z µ c / η,..., / ηd. Proposition The D-norm corresponding to the generator Z defined in 2.7 is given 62

69 by E c /c /c max xi Zi x,..., xd, i d D x 0 Rd. 2.8 If η,... ηd in the preceding result are independent, i.e., if the corresponding D-norm c c is k k, then Proposition c that Z µc / η,..., / ηd generates the P2.4. implies d /c. This was already observed in Proposition.2.. logistic norm kxk/c i xi Proof. Recall that by Fubini s theorem Z EY P Y > t dt, 0 if Y is an integrable rv with Y 0 a.s. We, consequently, obtain for x > 0 Rd xi E max xi Zi E max c i d i d ηi µc Z xi P max t dt c i d ηi µc 0 Z xi P max t dt c i d ηi µc 0 Z xi P t, i d dt µc 0 ηi c Z t P, i d dt µc 0 ηi c xi 63

70 exp µ c 0 t/x /c,..., t/x d /c dt D x /c exp µ c 0 t /c,..., x/c d D dt x /c µ,..., x/c d c exp dt c D 0 t /c by the substitution t x /c,..., x/c d c t. D The integral 0 exp /t /c dt equals by Fubini s theorem EY, where Y follows a Fréchet distribution with parameter /c. It was shown in 2.6 that EY µ c, which completes the proof. Iterating the Sequence of Generators Taking this new D-norm in 2.8 as the initial D-norm and proceeding as before leads to the D-norm x c 2 x,..., x d D 2 : /c2,..., x /c2 d, x 0 D Rd. We can iterate this problem and obtain in the n-th step x c n x,..., x d D n : /cn,..., x /cn d, x 0 D Rd. The question suggests itself: Does this sequence of D-norms converge? 64

71 Note: If we choose D, then we obtain for x 0 R d x /c,..., c c x/c d max i d x/c i max x i x,..., x d. i d D The conjecture might, therefore, occur that the sequence of D-norms converges to the sup-norm, if it converges. Recall that D for an arbitrary D-norm and that c 0,. Consequently, we obtain by Lemma.. and, hence, x x,..., x d D n /cn x /cn,..., x /cn d c n i,..., x /cn d x /cn i d c n cn n x,... x d, x 0 R d, x,..., x d D n n x,... x d. D 65

72 66

73 Chapter 3 The Functional D-Norm 3. Introduction Some Basic Definitions By C [0, ] : {g : [0, ] R, g is continuous} we denote the set of continuous functions from the interval [0, ] to the real line. By E [0, ] we denote the set of those bounded functions f : [0, ] R with only a finite number of discontinuities. Note that E [0, ] is a linear space: If f, f 2 E [0, ] and x, x 2 R, then x f + x 2 f 2 E [0, ] as well. Let now Z Z t t [0,] be a stochastic process on [0, ], i.e., Z t is a rv for each t [0, ]. We require that each sample path of Z t t [0,] is a continuous function on [0, ], Z C [0, ], for short. We also require that Z t 0, EZ t, t [0, ], 67

74 and Then E sup Z t <. 0 t f D : E sup ft Z t, f E [0, ], 0 t defines a norm on E [0, ]: We, obviously, have f D 0 and f D E sup ft Z t E sup ft sup Z t 0 t t [0,] t [0,] sup ft t [0,] E sup Z t <. t [0,] Let f D 0. We want to show that f 0. Suppose that there exists t 0 [0, ] with ft 0 0, then 0 f D E sup ft Z t t [0,] E ft 0 Z t0 ft 0 EZ t0 ft 0 > 0, 68

75 which is a clear contradiction. We, thus, have established the implication f D 0 f 0. The reverse implication is obvious. Homogeneity is obvious as well: We have for f E[0, ] and λ R λf D E sup λft Z t 0 t E λ sup ft Z t 0 t λ E sup ft Z t 0 t λ f D. The triangle inequality for D follows from the triangle inequality for real numbers x + y x + y, x, y R: f + f 2 D E sup f + f 2 Z t 0 t E sup f Z t + f 2 Z t 0 t E sup f Z t + sup f 2 Z t 0 t 0 t E sup f Z t 0 t + E sup f 2 Z t 0 t 69

76 f D + f 2 D, f, f 2 E [0, ]. Measurability of Integrand Note that ftz t t [0,] is for each f E [0, ] a stochastic process whose sample paths have only a finite number of discontinuities, namely those of the function f. We, therefore, can find a sequence of increasing index sets T n {t,... t n }, n N, such that sup ft Z t lim max ft t [0,] n i Z ti. i n As max ft i Z ti is for each n N a rv, the limit of this sequence, i.e., sup ft Z t, i n t [0,] is a rv as well. We, therefore, can compute its expectation, which is finite by the bound sup ft Z t : fz t [0,] sup ft sup Z t t [0,] t [0,] f Z and taking expectations. Recall that each function f E[0, ] is by the definition of E[0, ] bounded. The process Z Z t t [0,] is again called generator of the D-norm D. 70

77 The Brown-Resnick Pro- Example of a Generator: cess A nice example of a generator process is the Brown-Resnick process Brown and Resnick 977 Z t : exp B t t, t [0, ], 2 where B : B t t [0,] is a standard Brownian motion on [0, ]. That is, B C[0, ], B 0 0 and the increments B t B s are independent and normal N0, t s distributed rv with mean zero and variance t s. As a consequence, each B t with t > 0 is N0, t- distributed. We, therefore, have Z t > 0, t [0, ], and, for t > 0, EZ t exp t EexpB t 2 exp t E exp t /2 B t 2 t /2 exp t expt /2 x 2 2π exp x t/2 2 dx 2π /2 2, 7 exp x2 /2 2 dx

78 as exp x t /2 2 /2 /2π /2 is the density of the normal Nt /2, -distribution. It is well known that for x 0 and, thus, E P sup Z t E t [0,] sup B t > x t [0,] E sup t [0,] exp P 2P B > x exp B t sup t [0,] exp P + 2EexpB < B t sup B t > x t [0,] sup B t > logx t [0,] P B > logx dx as expb is standard lognormal distributed, with expectation exp/2. The computation of the corresponding D-norm is, however, not obvious JF3 Lec7.pdf dx dx 72

79 Bounds for the Functional D-Norm Lemma 3... Each functional D-norm is equivalent with the sup-norm k k, precisely, kf k kf kd kf k kkd, f E [0, ]. Proof. Let Z Zt t [0,] be a generator of k kd. We have for each t0 [0, ] and f E [0, ] f t0 E f t0 Zt0! E sup f t Zt t [0,] kf kd E kf k kzk kf k kkd, which proves the lemma. The Functional Lp-Norm is not a D-Norm Different to the multivariate case, the functional logistic norm is not a functional Dnorm. 73

80 Corollary 3... Each p-norm kf kp : R 0 /p f t dt with p [, is not a D-norm. p Proof. Choose ε 0, and put fε : [0,ε] E [0, ]. Then kfε k > ε/p kfε kp. The p-norm, therefore, does not satisfy the first inequality in the preceding result. A Functional Version of Takahashi s Theorem The next consequence of Lemma 3.. is obvious. This is a functional version of Takahashi s Theorem.3. for k k. Note that there cannot exist an extension to the functional case with k k, as this is not a functional D-norm by the preceding result. Corollary A functional D-norm k kd is the sup-norm k k iff kkd. 3.2 Generalized Pareto Processes Defining a Simple Generalized Pareto Process Let Z Zt t [0,] be the generator of a functional D-norm k kd with the additional property Zt c, t [0, ], 74 3.

81 for some constant c. For each functional D-norm there exists a generator with this additional property, see de Haan and Ferreira 2006, equation This might be viewed as a funcitonal analogue of the Normed Generators Theorem.7.2. Let U be a rv that is uniformly distributed on 0, and which is independent of Z. Put V : V t t [0,] : U Z t t [0,] : Z. 3.2 U Repeating the arguments in equation 2.3 we obtain for g E [0, ] with gt c, t [0, ], P V g 3.3 P U Z g U Z t, t [0, ] gt P [0,c] [0,] P [0,c] [0,] P U P [0,c] [0,] [0,c] [0,] z t gt U sup sup t [0,] t [0,], t [0, ] z t gt U sup t [0,] z t gt P Z dz t t [0,] P Z dz t t [0,] z t gt P Z dz t t [0,] 75 P Z dz t t [0,]

82 Zt t [0,] gt g, E! sup D i.e., the functional df of the process V is in its upper tail given by k/gkd. We have, moreover, x c, t [0, ], P Vt x, x i.e. each marginal df of the process V is in its upper tail equal to the standard Pareto distribution. We, therefore, call the process V simple generalized Pareto process; see Ferreira and de Haan 204 and Dombry and Ribatet 205. Survival Function of a Simple Generalized Pareto Process The following result extends the survival function of a multivariate GPD as in equation 2.4 to simple generalized Pareto processes. Proposition Let Z Zt t [0,] be the generator of a functional D-norm k kd with the additional property kzk c for some constant c. Then we obtain for g E [0, ] 76

83 with gt c, t [0, ], P V g P V > g E inf t [0,] Zt gt. Proof. Repeating the arguments in equation 3.3, we obtain P V > g Z zt P U<, t [0, ] P Z dzt t [0,] [0,] gt [0,c] Z zt P U inf P Z dzt t [0,] [0,] t [0,] gt Z[0,c] zt P Z dzt t [0,] inf [0,] t [0,] gt [0,c] Zt E inf. t [0,] gt Excursion stability of a Generalized Pareto Process Corollary We obtain under the conditions of Lemma 3.2. and the additional condition 77

84 E inf t [0,] Zt > 0 P V xg V g, x x. Proof. We have P V xg V g P V xg, V g P V g P V xg P V g Zt E inf t [0,] xgt. x Zt E inf gt t [0,] The conditional excursion probability P V xg V g /x, x, does not depend on g. We, therefore, call the process V excursion stable. 78

85 Sojourn Time of a Stochastic Process The time, which the process V V t t [0,] spends above the function g E [0, ] g c, is called its sojourn time above g, denoted by ST g From Fubini s theorem we obtain E ST g E gt, V t dt. 0 gt, V t dt E gt, V t dt P V t > gt dt gt dt. Recall that P V t x /x, x c, t [0, ]. By choosing the constant function gt : s c, we obtain for the expected sojourn time of the process V above the constant s EST s E s, V t dt s. 0 79

86 This implies independent of s c. EST s EST s ST s > 0 P ST s 0 /s P V t s, t [0, ], D 3.3 Max-Stable Processes Introducing Max-Stable Processes Let V, V 2,... be a sequence of independent copies of V Z/U, where the generator Z satisfies the additional boundedness condition 3.. We obtain for g E [0, ], g > 0, P n max P n P i V i g i n V i ng, i n V i ng 80

87 P V ng n n ng D exp n g, D where the mathematical operations max i n V n i, etc. are taken componentwise. The question now occurs: Is there a stochastic process ξ ξ t t [0,] on [0, ] with P ξ g exp g, g E [0, ], g > 0? D If ξ actually exists: Does it have continuous sample paths? If such ξ exists, it is a max-stable process: Let ξ, ξ 2,... be a sequence of independent copies of the process ξ. Then we obtain for g E[0, ], g > 0, and n N P n max i n ξi g P max i n ξi ng ξ i ng, i n P n P i ξ i ng P ξ ng n exp n ng D 8

88 exp n P ξ g. n ng D For the existence of such processes see Theorem

89 Chapter 4 Tutorial: D-Norms & Multivariate Extremes 4. Univariate Extreme Value Theory Let X be R-valued random variable rv and suppose that we are only interested in large values of X, where we call a realization of X large, if it exceeds a given high threshold t R. In this case we choose the data window A t, or, better adapted to our purposes, we put t R on a linear scale and define A n a n t + b n, for some norming constants a n > 0, b n R. We are, therefore, only interested in values of X A n. Denote by F the distribution function df of X. We obtain for s 0 P {X a n t + s + b n X > a n t + b n } F a nt + s + b n, F a n t + b n 83

90 thus facing the problem: What is the limiting behavior of F a n t + s + b n F a n t + b n n? 4. Extreme Value Distributions Let X, X 2,... be independent copies of X. Suppose that there exist constants a n > 0, b n R such that for x R max i n X i b n P x F n a n x + b n n Gx a n for some non degenerate limiting df G. Then we say that F belongs to the domain of attraction of G, denoted by F DG. In this case we deduce from the Taylor expansion log + ε ε + Oε 2 for ε 0 the equivalence F n a n x + b n n Gx n log F a n x + b n n loggx n F a n x + b n n loggx if 0 < Gx, and hence, F a n t + s + b n F a n t + b n n loggt + s loggt 4.2 if 0 < Gt <. 84

91 By the meanwhile classical article by Gnedenko 943 see also de Haan 975 and Galambos 987 we know that F DG only if G {G α : α R}, with G α x { exp x α, x 0,, x > 0, for α > 0, and G α x { 0, x 0, exp x α, x > 0, G 0 x : exp e x, x R, for α < 0 being the family of reverse Weibull, Fréchet and the Gumbel distribution. Note that G x expx, x 0, is the standard inverse exponential df. Max-Stability of Extreme Value Distributions The characteristic property of the class of the extreme value distributions EVD {G α : α R} is their max-stability, i.e., for each α R and each n N there exist constants a n > 0, b n R, depending on α, such that G n a n x + b n Gx, x R. 4.3 For Gx expx, x 0, for example, we have a n /n, b n 0, n N: x x n G n exp expx Gx. n n 85

92 Let η, η 2,... be independent copies of a rv η that follows the df G α. In terms of rv, equation 4.3 means max i n η i b n P x P η x, x R. a n This is the reason, why G α is called a max-stable df, and the set {G α : α R} collects all univariate max-stable distributions which are non degenerate, i.e., they are not concentrated in one point in R see, e.g., Galambos 987, Theorem Generalized Pareto Distributions If we assume that F DG α, we obtain from 4.2 that X bn P t + s X b n > t a n a n n F a nt + s + b n n F a n t + b n n logg αt + s logg α t { H α + s t, if α 0, s 0, H 0 s, if α 0. provided 0 < G α t <. The family H α s : + logg α s, 0 < G α s <, 86

93 s α, s 0, if α > 0, s α, s, if α < 0, exp s, s 0, if α 0, of df parameterized by α R is the class of univariate generalized Pareto df GPD coming along with the family of EVD. Notice that H α with α < 0 is a Pareto distribution, H is the uniform distribution on, 0, and H 0 is the standard exponential distribution. The preceding considerations are the reason, why random exceedances above a high threshold are typically modelled as iid observations coming from a univariate GPD. It was, for example, first observed by van Dantzig 960 that floods, which exceed some high threshold, follow approximately an exponential df. Consequence: Suppose that your data are realizations from iid observations, whose common df is in the domain of attraction of an extreme value df. Almost every textbook df satisfies this condition. Then the approximation of exceedances above high thresholds by a GPD is, consequently, a straightforward option and typically used in risk assessment. 4.2 Multivariate Extreme Value Distributions In complete accordance with the univariate case we call a df G on R d max-stable, if for every n N there exists vectors a n > 0, b n R d such that G n a n x + b n Gx, x R d

94 All operations on vectors such as addition, multiplication etc. are always meant componentwise. The preceding equation can again be formulated in terms of componentwise maxima of independent copies η, η 2,... of a rv η η,..., ηd that realizes in Rd, and which follows the df G: max i n η i bn P x P η x, x Rd. an Note that also max is taken componentwise as well as division. Different to the univariate case, the class of multivariate max-stable distributions or multivariate extreme value distributions EVD is no longer a parametric one, indexed by some α. This is obviously necessary for the univariate margins of G. Instead, a nonparametric part occurs, which can be best described in terms of D-norms. What is a D-Norm? Definition A norm k kd on Rd is a D-norm, if there exists a rv Z Z,..., Zd with Zi 0, EZi, i d, such that kxkd E max xi Zi, i d x x,..., xd Rd. In this case the rv Z is called generator of k kd. 88