Transformations of Copulas and Measures of Concordance

Transcription

1 Transformations of Copulas an Measures of Concorance D I S S E R T A T I O N zur Erlangung es akaemischen Graes Doctor rerum naturalium (Dr. rer. nat.) vorgelegt er Fakultät Mathematik un Naturwissenschaften er Technischen Universität Dresen von Dipl. Math. Sebastian Fuchs geboren am in Pirna Gutachter: Prof. Dr. Klaus D. Schmit Technische Universität Dresen Prof. Dr. Fabrizio Durante Free University of Bozen-Bolzano Eingereicht am: 3. Juli 205 Tag er Disputation: 27. November 205 Diese Dissertation wure in er Zeit von März 200 bis Juli 205 am Institut für Mathematische Stochastik angefertigt.

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3 Contents Introuction iii Copulas. Basic Properties an Results The Bivariate Case A Group of Transformations 9 2. Elementary Transformations on Copulas A Group of Transformations on Copulas Subgroups an Generators Total Reflection an Survival Copula Symmetry an Orer Preserving Subgroups Invariance of Copulas The Bivariate Case A Biconvex Form An Isomorphic Group of Transformations Copula Measures A Biconvex Form on Copulas The Biconvex Form an the Group of Transformations The Bivariate Case Measures of Concorance Basic Properties an Results Generator Inuce Measures of Concorance Copula Inuce Measures of Concorance The Bivariate Case A Result Beyon Copulas The Group of Transformations The Biconvex Form Measures of Concorance Conclusion an Outlook 3 i

4 Contents A The Map η K 35 A. Basic Properties an Results B Difference Operator 37 B. Partial Difference Operator B.2 Difference Operator Bibliography 43 List of Symbols 49 Inex 5 ii

5 Introuction In recent years, the theory of copulas has become a rapily eveloping fiel of probability theory an copulas are nowaays a wiely use tool in statistics with applications in insurance, finance, hyrology, etc. The term copula (lat. to connect, to join) has its origin in the work of Abe Sklar from 959, an enotes a imensional ( 2) istribution function on the unit hypercube with uniform margins. Sklar showe that, for every imensional istribution function F with margins F,..., F, there exists at least a copula C satisfying F (x,..., x ) C (F (x ),..., F (x )) (see Sklar [959, 973]). In the case of continuous margins C is unique. This theorem allows to stuy the epenence structure of a istribution function expresse by the copula without consiering the margins. The principal aim of the present work is, first, to introuce an investigate a group Γ of transformations mapping the collection C of all imensional copulas into itself an, secon, to apply the group Γ to efine, stuy an generate measures of concorance for multivariate copulas of a fixe imension. Transformations of copulas (better to say transforme copulas) are nearly as ol as the theory of copulas itself (see e.g. Sklar [973, p. 457] an Wolff [980, p.79]), an over time they became more an more iversifie; see e.g. Mikusiński et al. [99], Li et al. [998], Klement et al. [2002], Durante & Sempi [2005], Morillas [2005], Nelsen [2006, Chapter 3], Siburg & Stoimenov [2008], Navarro & Spizzichino [200], Dolati et al. [204] an Durante et al. [204b]. Some transformations of copulas, however, are of particular interest since they prouce new copulas from a given one: This inclues for example the transformations which turn a copula into a permute copula, a reflecte copula or its survival copula. The survival copula seems to be one of the first consiere copulas which emerge from a given one; see Wolff [980, p.79]. These three types of copulas mainly acquire its prominence in connection with measures of epenence; see e.g. Schweizer & Wolff [98], Scarsini [984a, 989] an Ewars et al. [2004, 2005] for the bivariate case, an for the multivariate case see e.g. Wolff [980] an Dolati & Úbea Flores [2006a]. Permute copulas play also a significant role in orer statistics (see e.g. Navarro & Spizzichino [200]), an the survival copula is of particular interest in connection with survival moels (see e.g. Bassan & Spizzichino [2005]). iii

6 Introuction Containing these three types of transformations, the group Γ is a realization of the hyperoctaheral group stuie among others by Young [930], Kerber [97] an Baake [984]. It is thus isomorphic to the well known group Γ of symmetries on the unit hypercube consiere for example by Ewars et al. [2004 & 2005], Taylor [2007, 2008 & 200] an Ewars & Taylor [2009] in connection with measures of concorance; see also Klement et al. [2002] an Nelsen [2006, Exercise 2.6] who use such symmetries to efine certain copulas such as reflecte copulas or the survival copula of a given copula. In the mi 970s, Schweizer an Wolff observe that the well known measures of concorance Spearman s rho an Kenall s tau possess a representation which epens solely on copulas; see Wolff [980] an Schweizer & Wolff [98]. Only a few years later, Scarsini [984a] obtaine a representation of Gini s gamma in terms of copulas. Further representations of common measures of concorance in terms of copulas as well as various multivariate generalizations were presente, e.g., in Nelsen [99, 998, 2002], Joe [997], Ewars et al. [2004, 2005], Nelsen & Úbea Flores [2005], Úbea Flores [2005], Dolati & Úbea Flores [2006a,b], Taylor [2007, 2008 & 200], Schmi et al. [200]. Scarsini [984a] further propose a first axiomatic efinition of a measure of concorance (for bivariate ranom vectors with continuous margins). Two ranom variables X an Y are sai to be concorant if: Greater values of X go with greater values of Y (Kruskal [958, p.88, lines 29 30]); similar formulations were use in Tchen [980] an Joe [990]. Scarsini realize that this concept epens on the corresponing copula alone, an reformulate it in terms of copulas. Thus, the value of a measure of concorance for bivariate ranom vectors epens solely on the corresponing copula. However, a first proper copula base efinition of a measure of concorance (for bivariate copulas an base on Scarsini s axioms) was introuce by Ewars et al. [2004] who use the group Γ of symmetries on the unit hypercube; see also Ewars et al. [2005] an Ewars & Taylor [2009]. Dolati & Úbea Flores [2006a] then propose a efinition of a measure of concorance for multivariate copulas of a fixe imension, an Taylor [2007, 2008, 200] introuce a efinition of a sequence of measures of concorance for copulas. Both efinitions are closely relate: Taylor s efinition involves the group Γ, an that of Dolati an Úbea Flores employs specific transformations in Γ. A quite general efinition of a measure of concorance for bivariate copulas was later propose by Fuchs & Schmit [204], an a generalization to a measure of concorance for multivariate copulas of a fixe imension was stuie in Fuchs [204]; both applie the group Γ instea of Γ. Schweizer an Wolff aitionally observe that Spearman s rho an Kenall s tau can even be written in terms of a linear functional C α [0,] 2 C(u, v) Q D (u, v) β where α an β are some fixe real numbers, D is a specific copula an Q D enotes the probability measure associate with D; see Wolff [980] an Schweizer & Wolff iv

7 [98]. Several years later, Nelsen [998] observe that also Gini s gamma possesses such a linear representation in terms of copulas. A more general result concerning copula inuce measures of concorance with linear structure was then presente in Ewars et al. [2004]. The authors showe that every copula D which is invariant with respect to all transformations in the group Γ inuces a measure of concorance. This result generates a variety of new measures of concorance incluing the examples Spearman s rho an Gini s gamma. In this context, we also refer to Ewars et al. [2005] an Ewars & Taylor [2009]. Dolati & Úbea Flores [2006a] an Taylor [2007] then showe that the mentione invariance conition remains sufficient also for the multivariate case. However, a slight moification of the consiere integral is require. The outline of this thesis is as follows: In Chapter we present a short an precise introuction to the theory of copulas an, in aition, we provie all tools which are necessary for the rest of this work. In Chapter 2 we introuce the group Γ of transformations mapping the collection of all copulas into itself. We present a systematic investigation of this group starting with the efinition of elementary transformations on the collection of all copulas. These elementary transformations turn out to be most helpful for the construction of the group. Since for the hyperoctaheral group no general subgroup structure is known, we thus restrict our consieration to specific subgroups of Γ for which we present minimal systems of generators. One subgroup of Γ is of particular importance since its! 2 elements are the only transformations in Γ which preserve the symmetry of a copula an the concorance orer between two copulas. We further confirm that the group contains a transformation which turns every copula into its survival copula. In aition, we stuy copulas which are invariant with respect to all transformations in a subgroup of Γ. For a systematic investigation of measures of concorance, it is appropriate to stuy the probability measure corresponing to a copula which we here call copula measure, an to introuce a bilinear form (better to say biconvex form) on C C to stuy the integration of a copula with respect to a copula measure. Both is one in Chapter 3. In aition, we investigate copula measures an the biconvex form with regar to the group Γ an invariant copulas. We also show that the transformations preserving the value of the biconvex form form a subgroup as well. Chapter 4 an Chapter 5 are eicate to measures of concorance. We here follow but slightly moify the efinition propose by Fuchs [204] an stuy measures of concorance for multivariate copulas of a fixe imension. First, we show that the transformations in Γ provie a simple test on copulas uner which every measure of concorance is equal to zero. Moreover, it turns out that the transformations which preserve the symmetry of a copula an the concorance orer between two copulas are also the only transformations in Γ which preserve the value of every measure of concorance. In aition, we present a generator base an, in particular, a copula base principle for the construction of measures of concorance. For the latter, v

8 Introuction we iscuss appropriate moifications of the consiere integral, an we provie a quite abstract characterization of copula inuce measures of concorance. As a consequence of this characterization, we then obtain a sufficient conition on a copula in terms of the transformations in Γ uner which this copula inuces a measure of concorance. This is the main result of Chapter 4. As a consequence an analogously to the results of Dolati & Úbea Flores [2006a] an Taylor [2007], it turns out that every copula which is invariant with respect to all transformations in Γ, an even every copula, inuces a measure of concorance. In Chapter 5 we then introuce a more complex tool which is intene to improve the mentione characterization of copula inuce measures of concorance an the main result of Chapter 4. To this en an contrary to Chapters to 4, we rop the consieration of a fixe imension, an introuce multivariate margins of copulas which turn out to be copulas as well. We then investigate them with regar to the group Γ of transformations an the biconvex form on copulas; however, we restrict ourselves to only those results which are essential for the comprehension. The last section of each Chapter, 2, 3 an 4 is eicate to the bivariate case. This is of interest since there exist various results that are solely vali for this case. In aition, the representation of some results can be improve or simplifie. In Chapter an Chapter 2 we eliberately avoi the use of results or even notions from measure an probability theory, an stuy the collection of all copulas an the group Γ of transformations from a purely analytical point of view. Thus, these two chapters may be consiere as a self containe part of this work. Acknowlegement A lot of thanks go to the members of the Institute of Mathematical Stochastics for their manifol support. In particular, I woul like to thank Dr. Georg Berschneier for his encouragement especially at the beginning of this work, an for proofreaing an earlier manuscript of this thesis. Especially, I woul like to express my gratitue to my supervisor Prof. Dr. Klaus D. Schmit who allowe me enough time for eveloping, elaborating an finishing this thesis. Thank you for the valuable comments an inspiring conversations. I enjoye your inexhaustible curiosity. Last, but not least, I woul like to thank Juith Fuchs for her unerstaning an patience. vi

9 Chapter Copulas Our aim in this first chapter is to present a short an precise introuction to the theory of copulas an, in this context, to provie all tools which are necessary for the rest of this work. In his Ph.D. thesis Maßstabinvariante Korrelationstheorie, publishe in 940, Wassily Hoeffing escribe a metho which transforms a bivariate istribution function into a istribution function with uniform margins which acts on the intervall [ 2, 2 ]2. The emerging function he calle normierte Summenfunktion (engl. translation: stanarise istribution function). Hoeffing aitionally observe that several known scale invariant measures of epenence, such as Spearman s rho, possess a representation which epens solely on this stanarise istribution function; see Hoeffing [940]. Unfortunately, Hoeffing s ieas passe unnotice an it took some time for their reiscovery: About 20 years after Hoeffing s research, Abe Sklar prove that, for every imensional ( 2) istribution function F with margins F,..., F, there exists at least a function C with uniform margins satisfying F (x,..., x ) C (F (x ),..., F (x )) (see Sklar [959, 973]). In the case of continuous margins C is unique. Sklar calle those functions copulas. In contrast to Hoeffing s stanarise istribution functions, however, copulas act on the unit hypercube. In the years after Sklar s observation, copulas ha mainly become part of the theory of probabilistic metric spaces; see e.g. Schweizer & Sklar [983]. As late as the mi 970s an inspire by a paper of Rényi who iscusse axioms a measure of epenence shoul satisfy (see Rényi [959]), Schweizer an Wolff then showe that several measures of epenence such as Spearman s rho an Kenall s tau possess a representation in terms of copulas (see Wolff [980] an Schweizer & Wolff [98]). Since then, the theory of copulas has evelope rapily an has become a wiely varie an noteworthy fiel of stochastics. A more etaile historical review was presente by Schweizer [99]; for a well iversifie introuction to the theory of copulas we refer to Nelsen [2006], Schmit [2006], Jaworski et al. [200] an Joe [205].

10 Chapter. Copulas This chapter is organize as follows: In Section. we consier the collection of all copulas which we investigate with regar to orer relations an greatest elements. We further introuce important examples an, as a first transformation of a copula, we efine the survival copula of a copula. The final section is eicate to the bivariate case. Let be an integer 2 which will be kept fixe throughout this work. For the sake of a concise notation, let I [0, ] an, for K {,..., }, we consier the vector value function η K I I I given by u k k {,..., }/K (η K (u, v)) k v k k K an put η k η {k}, k {,..., }. Useful technical results relate to η K can be foun in Appenix A. Moreover, for k {,..., }, we enote by e k R the k th imensional unit vector an, for K {,..., }, let J K R be the matrix given by J K e k e k k K Then J is the matrix with entries 0, an J {,...,} coincies with the ientity matrix. For h I an a function f I R, we further efine the ifference operator by letting ( h (f))(u) ( ) K {,...,} ( ) K f(u + J K h) for all u [0, h]. Necessary efinitions an results concerning the ifference operator can be foun in Appenix B. By the imension of a function f I R we enote the imension of its arguments. Finally, for m N, we put N(m) {n N m n}.. Basic Properties an Results In this first section we consier the collection of all copulas which we investigate with regar to pointwise an concorance orer relation an greatest elements. We first introuce the two main an well known copulas: the Fréchet Hoeffing upper boun M an the prouct copula Π. We show that the Fréchet Hoeffing upper boun is the greatest element (with regar to pointwise an concorance orer relation) an, in aition, an extremal point of the collection of all copulas. As a first transformation of a copula, we efine the survival copula of a copula which turns out to be a copula as well. We further introuce a very useful collection of copulas which we call istortions of Π. Those copulas will be extremely helpful throughout this work. We conclue this section with some well known facts concerning continuity an convergence. 2

11 .. Basic Properties an Results A function f I R is sai to be monotone if the inequality ( h (f))(u) 0 hols for all h I an all u [0, h], groune if the ientity f(η i (u, 0)) 0 hols for all u I an all i {,..., }, an f is sai to possess uniform univariate margins if the ientity f(η i (, u)) u i hols for all u I an all i {,..., }. A copula is a function C I R satisfying the following conitions: (i) C is monotone. (ii) C is groune. (iii) C possesses uniform univariate margins. We enote by M the collection of all functions I R an by C the collection of all copulas. Then M is a vector space uner the coorinatewise efine linear operations, an C is a convex subset of M. For C, D M, we write C D if C(u) D(u) for all u I. Then is an orer relation an is calle pointwise orer relation, an the pair (M, ) is an orere vector space. The following both well known examples will play an important role throughout this work:.. Examples. () Fréchet Hoeffing upper boun: The function M I R given by M(u) min{u i i {,..., }} is monotone, groune an possesses uniform univariate margins, an thus, M is a copula. Moreover, the inequality C M hols for every copula C C ; see Nelsen [2006, p.47]. Inee, for all h I an all u [0, h], we have ( h (M))(u) max { min {u i + h i, i {,..., }} max {u i, i {,..., }}, 0} 0 (see Nelsen [2006, Exercise 2.35]) which proves (i); conitions (ii) an (iii) are obvious. 3

12 Chapter. Copulas (2) Prouct copula: The function Π I R given by Π(u) u i i is monotone, groune an possesses uniform univariate margins, an thus, Π is a copula. Inee, for all h I an all u [0, h], we have ( h (Π))(u) Π(h) 0 (see Nelsen [2006, Exercise 2.35]) which proves (i); conitions (ii) an (iii) are obvious. The next result is evient from Example.. ():..2 Corollary. () M is the greatest element of (C, ). (2) M is an extremal point of C. In this work we omit a etaile iscussion about extremal points of C. Those copulas are still subject of research, an there seem to be two ifferent approaches to eal with extremal points: shuffles of M (see e.g. Mikusiński et al. [992] an Durante et al. [2009]) an copulas with hairpin support (see e.g. Kamiński et al. [990] an Durante et al. [204a]). As a very first transformation of a copula C C, we efine the function Ĉ I R by letting Ĉ(u) ( ) ( ) K C(η K ( u, )) K {,...,} Ĉ is calle the survival copula of C; this efinition is in accorance with that of Durante & Sempi [200]. We resume the iscussion of copulas M an Π initiate in Examples..:..3 Examples. () The copula M satisfies M M; see Dolati & Úbea-Flores [2006a, p.48]. (2) The copula Π satisfies Π Π; see Dolati & Úbea-Flores [2006a, p.48]. The survival copula of a copula is a copula as well. This result is well known; see also Theorem below...4 Proposition. For every C C we have Ĉ C. For C, D C, we write C c D if C D an Ĉ D. Then c is an orer relation on C an is calle concorance orer relation. Moreover, Corollary..2 an Example..3 () imply that the inequality C c M hols for every copula C C. We thus obtain the following result:..5 Corollary. M is the greatest element of (C, c ). 4

13 .. Basic Properties an Results The concorance orer relation for copulas was introuce by Scarsini [984a] for the bivariate case, an (to the best of our knowlege) by Nelsen [2002] for the general case. We further introuce a very useful collection of copulas. A function C I R is calle istortion of Π if it satisfies C Π + Π g for some vector g (g i ) i {,...,} of functions g i I I, i {,..., }, such that (i) the inequality i h i + i (g i (u i + h i ) g i (u i )) 0 hols for all h I an all u [0, h], (ii) the ientity g i (0) 0 hols for all i {,..., }, an (iii) the ientity g i () 0 g j () hols for some i, j {,..., } with i j. We obtain the following result:..6 Lemma. Every istortion of Π is a copula. Proof. Let C I R be a istortion of Π. Then C Π + Π g for some vector g of functions satisfying conitions (i), (ii) an (iii). First, (ii) an (iii) imply that C is groune an possesses uniform univariate margins. We now prove that C is monotone. To this en, consier h I an u [0, h]. Linearity of the ifference operator (Corollary B.2.), Lemma B.2.3, Example.. (2) an the ientity ( g(u+h) g(u) (Π))(g(u)) ( ) ( ) K Π(g(u) + J K (g(u + h) g(u))) K {,...,} yiel ( ) ( ) K Π(g(u) + g(u + J K h) g(u)) K {,...,} ( ) ( ) K g k (u k + h k ) g k (u k ) K {,...,} k K k K i (g i (u i + h i ) g i (u i )) ( h (C))(u) ( h (Π + Π g))(u) ( h (Π))(u) + ( h (Π g))(u) ( h (Π))(u) + ( g(u+h) g(u) (Π))(g(u)) 0 i h i + i (g i (u i + h i ) g i (u i )) This proves the assertion. 5

14 Chapter. Copulas The introuce collection of istortions of Π inclues the well known Farlie Gumbel Morgenstern family of copulas (see Nelsen [2006, p.77]). In aition, similar istortions of the prouct copula were stuie by Dolati & Úbea Flores [2006a, 2006b]. For the sake of simplicity, for all upcoming examples of istortions of Π, we omit any further etaile information about their affiliation to a specific family of copulas. The following example is a istortion of Π:..7 Example. For m,..., m N, the function E I R given by E(u) Π(u) + i u m i i ( u i ) is a istortion of Π an hence a copula. Inee, for all h I, all u [0, h] an all i {,..., } with m i, we first obtain ( u i ) (u i + h i ) Moreover, for all h I, all u [0, h] an all i {,..., } with m i N(2), we have m i l0 u m i l i (u i + h i ) l m i l0 m i l0 m i j0 u m i l i l j0 l j0 ( l j ) um i j i ( l j ) ul j i h j i m i ( l lj j ) um i j i h j i m i ( m i j0 j + ) um i j i h j i h j i which implies m i ( u i ) j0 m i ( u i ) j0 ( u i ) u i m i 2 j0 ( m i j + ) um i j i h j i (u i + h i ) m i u m i j i (u i + h i ) j (u i + h i ) m i u m i 2 j i (u i + h i ) j + ( u i )(u i + h i ) mi (u i + h i ) m i m i 2 ( u i ) u i u m i 2 j i + ( u i )(u i + h i ) mi h i (u i + h i ) m i j0 u i ( u m i i ) + ( u i h i )(u i + h i ) m i an maximization of both terms then yiels u i ( u m i i ) + ( u i h i )(u i + h i ) m i ( m i ) ( m i m i m i m i m i ) + ( )( m m i i ) m i m i 6

15 .. Basic Properties an Results Thus i h i + ( m i ) m i ( m i m i ) + m i ( m i m i ) m i m i + m i ( m i m i ) m i + m i m i m i m2 i m 2 i i i i i i i [ + 0 m i ((u i + h i ) m i ( u i h i ) u m i i ( u i )) h i + h i + h i + h i + h i + i i i i i i (( u i )(u i + h i ) m i ( u i )u m i i h i (u i + h i ) m i ) m i (( u i ) ( m i j ) um i j i j0 m i (( u i ) j m i (( u i ) j0 m i h i (( u i ) m i (( u i ) j0 h j i ( u i)u m i i h i (u i + h i ) m i ) ( m i j ) um i j i h j i h i(u i + h i ) m i ) ( m i j + ) um i (j+) i h j+ i h i (u i + h i ) m i ) j0 ( m i j + ) um i j i h j i (u i + h i ) m i ) ( m i j + ) um i j i h j i (u i + h i ) m i )] which proves (i); conitions (ii) an (iii) are obvious. We conclue this section with some well known facts concerning continuity an convergence. i..8 Lemma. () Copulas are Lipschitz continuous with a uniform Lipschitz constant. (2) The collection C is equicontinuous. Proof. Assertion () was proven in Nelsen [2006, Theorem ], an (2) then follows from Lojasiewicz [988, p.49]. The final result of this section asserts that, for a sequence of copulas converging to a copula, pointwise convergence is the same as uniform convergence:..9 Lemma. Let {C n } n N C an C C. Then the following are equivalent: (a) {C n } n N converges pointwise to C. (b) {C n } n N converges uniformly to C. h i 7

16 Chapter. Copulas Proof. That (a) implies (b) follows from Lojasiewicz [988, Theorem 3.2.4] an the facts that I is compact an C is equicontinuous (Lemma..8). For the rest of this work, we thus restrict ourselves to uniform convergence of sequences of copulas..2 The Bivariate Case The final section of this chapter is eicate to the case 2. This is of interest since there exist various results about copulas that are solely vali for the bivariate case. In aition, the representation of some results can be improve or simplifie. We first show that, for 2, pointwise orer is the same as concorance orer. Further, we introuce another important an well known copula: the Fréchet Hoeffing lower boun W, which turns out to be the least element an, in aition, an extremal point of the collection of all copulas. The first characteristic we want to point out is the fact that, for 2, pointwise orer is the same as concorance orer. This result is well known (see e.g. Taylor [2007, p.794]), an its proof is straightforwar..2. Corollary. Let C, D C 2. Then the following are equivalent: (a) C D. (b) C c D. Besies the Fréchet Hoeffing upper boun an the prouct copula we consier another important copula:.2.2 Example. Fréchet Hoeffing lower boun: The function W I 2 R given by W (u, v) max{u + v, 0} is 2 monotone, groune an possesses uniform univariate margins, an thus, W is a copula; see Nelsen [2006, p.] or Example Moreover, the inequality W C hols for every copula C C 2 ; see Nelsen [2006, p.]. The following corollary is evient from Example.2.2 an extens the results given in Corollary..2 for the case 2; see also Darsow et al. [992, p. 604 & p.626ff]:.2.3 Corollary. () M is the greatest element of (C 2, ). (2) W is the least element of (C 2, ). (3) M an W are extremal points of C 2. 8

17 Chapter 2 A Group of Transformations Our aim in this chapter is to introuce an investigate a group Γ of transformations mapping the collection of all copulas into itself. The transformations in Γ possess a simple probabilistic interpretation: Let (Ω, F, P ) be a probability space an let X Ω R be a ranom vector whose margins X l, l {,..., }, are continuous. We enote by F X its joint istribution function an by F Xl the istribution function of X l, l {,..., }. Sklar s Theorem then asserts that there exists some unique copula C X satisfying F X (x,..., x ) C X (F X (x ),..., F X (x )) (see Sklar [959, 973]); analogously there exist unique copulas for the ranom vector (X,..., X i+, X i,..., X ) where the coorinates i an i + are interchange, for (X,..., X k,..., X ) where the kth coorinate changes sign, an for the ranom vector ( X, X 2,..., X ) where all coorinates change sign. The corresponing copulas are then connecte as follows C (X,...,X i+,x i,...,x )(u) C X (η {i,i+} (u, u i+ e i + u i e i+ )) C (X,..., X k,...,x )(u) C X (η k (u, )) C X (η k (u, u)) C ( X, X 2,..., X )(u) ( ) ( ) K C X (η K ( u, )) K {,...,} Thus, C (X,...,X i+,x i,...,x ) equals a permute copula of C X, C (X,..., X k,...,x ) a reflecte copula of C X, an C ( X, X 2,..., X ) equals the survival copula of C X. In fact, the obtaine copulas are transformations of C X. Instea of stuying these three types of copulas, we here follow the approach of Graf [2008] who, for the bivariate case, iscusse the corresponing transformations on its own, an thus, transformations mapping the collection of all copulas into itself; see also Ewars & Taylor [2009], Sachs [20], Neumann [202] an Fuchs & Schmit [204]. For the multivariate case, we refer to Taylor [2008, 200] an Fuchs [204]: We hence efine the maps π i,i+, ν k, τ C C by letting (π i,i+ (C))(u) C(η {i,i+} (u, u i+ e i + u i e i+ )) (ν k (C))(u) C(η k (u, )) C(η k (u, u)) 9

18 Chapter 2. A Group of Transformations an we efine τ to be the transformation which turns every copula into its survival copula. π i,i+ is calle a transposition, ν k is calle a partial reflection an τ is calle total reflection. The introuce transformations equippe with the composition form a group Γ consisting of! 2 elements; for the bivariate case see Neumann [202] an Fuchs & Schmit [204], an for the multivariate case we refer to Fuchs [204]. The group Γ is a realization of the hyperoctaheral group stuie for example by Young [930], Kerber [97] an Baake [984]. Copulas which are invariant with respect to all transformations in a subgroup of Γ play an important role in connection with measures of concorance; see e.g. Chapter 4 an Chapter 5 below. In this context, some subgroups of Γ such as the subgroup Γ π containing all transpositions, Γ ν containing all partial reflections (an hence the total reflection), an Γ τ containing the total reflection seem to be of particular interest. In literature, invariance with respect to all transformations in Γ π is in accorance with symmetry or exchangeability (see e.g. Nelsen [2006], Liebscher [2008], Durante & Papini [2009] an Navarro & Spizzichino [200]), invariance with respect to all transformations in Γ ν is in accorance with joint symmetry (see e.g. Nelsen [993, 2006]), an invariance with respect to all transformations in Γ τ is in accorance with reflection symmetry or raial symmetry (see e.g. Joe [997], Nelsen [993, 2006] an Dehgani et al. [203]). This chapter is organize as follows: In Section 2. we introuce transpositions an partial reflections as elementary transformations mapping the collection of all copulas into itself. In Section 2.2 we then construct the smallest group Γ containing all transpositions an partial reflections. Thus, Γ is a group of transformations mapping the collection of all copulas into itself. Since for the hyperoctaheral group no general subgroup structure is known, we thus restrict our consieration to specific subgroups of Γ for which we present minimal systems of generators (Section 2.3). The subgroup containing all transpositions an the total reflection is of particular interest since its 2! elements are the only transformations in Γ which preserve the symmetry of a copula an the concorance orer between two copulas. We further show that the transformations preserving orinary orer between two copulas form a subgroup as well, an we confirm that Γ contains a transformation which turns every copula into its survival copula (Sections 2.4 & 2.5). In Section 2.6 we stuy copulas which are invariant with respect to all transformations in a subset Λ of Γ. Those copulas we call Λ invariant. For every subset Λ of Γ, we hence obtain a collection of Λ invariant copulas which we investigate with respect to greatest elements. We further introuce the arithmetic mean of a copula with respect to a subset Λ of Γ which turns out to be a very useful tool. The final section is eicate to the case 2. The present chapter contains results an, in aition, several formulations of the paper Fuchs [204]. This inclues the construction of the group Γ in Sections 2. & 2.2, the introuction of several subgroups in Section 2.3 (without minimal systems of generators), the relation between the total reflection an the survival copula in Section 2.4, an the iscussion of subgroups preserving certain properties in Section

19 2.. Elementary Transformations on Copulas 2. Elementary Transformations on Copulas In this first section we introuce elementary transformations mapping the collection of all copulas into itself. All results an, in aition, several formulations of the present section are ue to Fuchs [204]. A map ϕ C M is sai to be a transformation. By the imension of a transformation we enote the imension of its arguments. Throughout this work we omit any specifications concerning the imension of a transformation or relate quantities since this is completely etermine by the imension of the copula the transformation is applie to. Let Φ enote the collection of all transformations mapping the collection of all copulas into itself, an efine the composition Φ Φ Φ by letting (ϕ ϕ 2 )(C) ϕ (ϕ 2 (C)). The composition is associative. The transformation ι Φ given by ι(c) C satisfies ι ϕ ϕ ϕ ι for all ϕ Φ an is calle the ientity on C. A transformation ϕ Φ is sai to be an involution if ϕ ϕ ι. Moreover, let Φ Φ be the set of all elements of Φ for which there exists an inverse element in Φ. For ϕ Φ, we then enote by ϕ the inverse element of ϕ. We obtain the following result: 2.. Lemma. (Φ, ) is a semigroup with neutral element ι. Moreover, (Φ, ) is a subgroup of Φ that contains all involutions. The term subgroup in Lemma 2.. means that (Φ, ) is a subsemigroup of (Φ, ) an a group. We now introuce two elementary transformations, transpositions an partial reflections, on which we will focus our attention in the subsequent sections: For i, j {,..., } with i j, we efine the map π i,j C M by letting (π i,j (C))(u) C(η {i,j} (u, u j e i + u i e j )) an, for k {,..., }, the map ν k C M by letting (ν k (C))(u) C(η k (u, )) C(η k (u, u)) π i,j is calle a transposition, an ν k is calle a partial reflection. It turns out that transpositions an partial reflections map the collection of all copulas into itself: 2..2 Theorem. Every transposition an every partial reflection is in Φ. Moreover, transpositions an partial reflections are involutions an hence in Φ. Proof. It is evient that every transposition of a copula is a copula. Now, consier C C an k {,..., }. We prove that ν k (C) is a copula as well. To this en, consier h I an u [0, h]. For all v [0, h k e k ], we first obtain

20 Chapter 2. A Group of Transformations (( k h k ν k )(C))(v) (ν k (C))(v + h k e k ) (ν k (C))(v) C(η k (v + h k e k, )) C(η k (v + h k e k, (v + h k e k ))) C(η k (v, )) + C(η k (v, v)) C(η k (v, )) C(η k (v, v h k e k )) C(η k (v, )) + C(η k (v, v)) C(η k (v, v)) C(η k (v, v h k e k )) C(η k (v, v h k e k ) + h k e k ) C(η k (v, v h k e k )) ( k h k (C))(η k (v, v h k e k )) Commutativity of the partial ifference operators (Corollary B..3), the ientity establishe before, the fact that η k (u, u h k e k ) [0, h] an the monotonicity of C then imply ( h (ν k (C)))(u) (( l h l ) (( k h k ν k )(C)))(u) l,l k ( ) ( ) L (( k h k ν k )(C))(u + J L h) L {,...,}/{k} ( ) ( ) L ( k h k (C))(η k (u + J L h, (u + J L h) h k e k )) L {,...,}/{k} ( ) ( ) L ( k h k (C))(η k (u + J L h, u h k e k )) L {,...,}/{k} ( ) ( ) L ( k h k (C))(η k (u, u h k e k ) + J L h) L {,...,}/{k} (( l h l ) ( k h k (C)))(η k (u, u h k e k )) l,l k ( h (C))(η k (u, u h k e k )) 0 which proves that ν k (C) is monotone; it is straightforwar to show that ν k (C) is groune an possesses uniform univariate margins. Further, it is evient that every transposition is an involution. Consier again C C an k {,..., }. Since C is groune, we obtain (ν k (ν k (C)))(u) (ν k (C))(η k (u, )) (ν k (C))(η k (u, u)) C(η k (η k (u, ), )) C(η k (η k (u, ), η k (u, ))) C(η k (η k (u, u), )) + C(η k (η k (u, u), η k (u, u))) C(η k (u, )) C(η k (u, 0)) C(η k (u, )) + C(u) C(u) for all u I. Therefore, every partial reflection is an involution as well. This proves the result. 2

21 2.2. A Group of Transformations on Copulas Theorem 2..2 states that every transposition an every partial reflection is in Φ. Thus, there exists a smallest subgroup of Φ containing all transpositions, a smallest subgroup of Φ containing all partial reflections, an a smallest subgroup of Φ containing all transpositions an all partial reflections. 2.2 A Group of Transformations on Copulas In the present section we construct the smallest subgroup Γ π of Φ containing all transpositions, the smallest subgroup Γ ν of Φ containing all partial reflections an the smallest subgroup Γ of Φ containing all transpositions an all partial reflections. Then Γ is a group of transformations mapping the collection of all copulas into itself, an Γ π an Γ ν are subgroups of Γ. This group Γ will be the main topic of this work. All results an, in aition, several formulations of the present section are ue to Fuchs [204]. For the composition of n N 0 transformations ϕ m Φ, m {,..., n}, we write n ϕ m m ι n 0 ϕ n n m ϕ m otherwise an, for N {,..., n} an a set of pairwise commuting ϕ m Φ, m N, we put ϕ m n ϕ m m N m Recall that the center of a group is the set of all those elements which commute with every element of the group, an the center is a subgroup. Let us first consier the smallest subgroup of Φ containing all transpositions. A transformation is calle a permutation if it can be expresse as a finite composition of transpositions. We enote by Γ π the set of all permutations. The following example will be neee for the proof of Theorem 2.2.2: 2.2. Example. Consier 3. For j {,..., }, the function E I R given by E(u) Π(u) + u j u i ( u i ) i,i j is a istortion of Π an hence a copula. Inee, for all h I an all u [0, h], we first obtain ( u i ) (u i + h i ) 3

22 Chapter 2. A Group of Transformations for all i {,..., }/{j} an hence i h i + (u j + h j u j ) i [ + 0 h i + h j i,i j ((u i + h i )( u i h i ) u i ( u i )) i,i j h i ( 2u i h i ) i,i j ( 2u i h i )] h i i which proves (i); conition (ii) is obvious an (iii) follows from Theorem. Γ π is the smallest subgroup of Φ containing all transpositions. Moreover, Γ π!. If 2, then this subgroup is commutative; if 3, then its center is trivial. Proof. Since every composition of two permutations is a permutation, Lemma 2.. implies that Γ π is a subsemigroup of Φ with neutral element ι. Now, let π Γ π. Then π can be expresse as a finite composition of transpositions. Let π be the composition of the same transpositions arrange in reverse orer, then π Γ π an Theorem 2..2 yiels π π ι π π Thus, Γ π is a subgroup of Φ. A permutation is a transformation that rearranges the coorinates of the arguments of a copula. Since there exist! possibilities to rearrange coorinates, we hence obtain the carinality of Γ π. For 2, we have Γ π {ι, π,2 } which shows that the elements of Γ π commute. Assume now that 3. To prove that the center of Γ π is trivial, consier π Γ π /{ι} an C C. Since π ι, there exist k, l {,..., } with k l such that the k th coorinate of the argument of C is the l th coorinate of the argument of π(c). Moreover, for j {,..., }, consier the copula E j (u) Π(u) + u j iscusse in Example We then have u i ( u i ) i,i j π(e k ) E l an for some m {,..., }/{k, l} (which exists since 3) we further obtain (π lm π π lm )(E k ) (π lm π)(e k ) π lm (E l ) E m E l π(e k ) This yiels π π lm π lm π. Thus, {ι} is the center of Γ π. Now, let us consier the smallest subgroup of Φ containing all partial reflections. A transformation is calle a reflection if it can be expresse as a finite composition of partial reflections. We enote by Γ ν the set of all reflections. 4

23 2.2. A Group of Transformations on Copulas Lemma. The ientity ν k ν l ν l ν k hols for all k, l {,..., }. Proof. Let C C an k, l {,..., } with k l. Then (ν k (ν l (C)))(u) (ν l (C))(η k (u, )) (ν l (C))(η k (u, u)) C(η l (η k (u, ), )) C(η l (η k (u, ), η k (u, ))) C(η l (η k (u, u), )) + C(η l (η k (u, u), η k (u, u))) C(η {k,l} (u, )) C(η {k,l} (u, e k + ( u l ) e l )) C(η {k,l} (u, ( u k ) e k + e l )) + C(η {k,l} (u, u)) for all u I, an hence ν k ν l ν l ν k. Since every partial reflection is an involution (Theorem 2..2), this proves the assertion. Thus, a reflection can be expresse as a finite composition of partial reflections containing every partial reflection at most once. The following example will be neee for the proof of Theorem 2.2.5: Example. The copula E(u) Π(u) + i u 2 i ( u i ) iscusse in Example..7 (m i 2 for all i {,..., }) satisfies π(e) E for all π Γ π an ν(e) ν(e) for all ν, ν Γ ν with ν ν Theorem. Γ ν is the smallest subgroup of Φ containing all partial reflections. Moreover, Γ ν 2. This subgroup is commutative an each of its elements is an involution. Proof. Since every composition of two reflections is a reflection, Lemma 2.. implies that Γ ν is a subsemigroup of Φ with neutral element ι. Moreover, since partial reflections commute an are involutions, it follows that in fact all reflections commute an are involutions. In particular, Γ ν is a subgroup of Φ. Since every reflection can be expresse as a finite composition of partial reflections containing every partial reflection at most once, the carinality of Γ ν is at most 2. That Γ ν 2 then follows from Example since the application of the reflections in Γ ν to E prouces 2 istinct copulas. Finally, let us consier the smallest subgroup of Φ containing all transpositions an all partial reflections. A transformation is calle a symmetry if it can be expresse as a composition of a permutation an a reflection. We enote by Γ the set of all symmetries. Note that every symmetry can be expresse as a finite composition of transpositions an partial reflections. 5

24 Chapter 2. A Group of Transformations We nee the following lemma which inclues results for the composition of a transposition an a partial reflection: Lemma. Let i, j {,..., } with i j an k {,..., }/{i, j}. Then () π i,j ν j ν i π i,j (2) π i,j ν i ν j π i,j (3) π i,j ν k ν k π i,j In particular, π i,j commutes neither with ν i nor with ν j. Proof. Let C C. Then we obtain (π i,j (ν j (C)))(u) (ν j (C))(η {i,j} (u, u j e i + u i e j )) C(η j (η {i,j} (u, u j e i + u i e j ), )) C(η j (η {i,j} (u, u j e i + u i e j ), η {i,j} (u, u j e i + u i e j ))) C(η {i,j} (u, u j e i + e j )) C(η {i,j} (u, u j e i + ( u i ) e j )) (π i,j (C))(η i (u, )) (π i,j (C))(η i (u, u)) (ν i (π i,j (C)))(u) for all u I. This proves (). Analogously, we obtain (2). Moreover (π i,j (ν k (C)))(u) (ν k (C))(η {i,j} (u, u j e i + u i e j )) C(η k (η {i,j} (u, u j e i + u i e j ), )) C(η k (η {i,j} (u, u j e i + u i e j ), η {i,j} (u, u j e i + u i e j ))) C(η {i,j,k} (u, u j e i + u i e j + e k )) C(η {i,j,k} (u, u j e i + u i e j + ( u k ) e k )) (π i,j (C))(η k (u, )) (π i,j (C))(η k (u, u)) (ν k (π i,j (C)))(u) for all u I. This proves (3). The present lemma comprises results for the composition of a permutation an a reflection, an it also shows that the intersection of the subgroups Γ π an Γ ν is trivial: Lemma. () For every π Γ π an every ν Γ ν there exists some π Γ π an some ν Γ ν such that π ν ν π (2) For every π Γ π an every ν Γ ν there exists some π Γ π an some ν Γ ν such that ν π π ν (3) Γ π Γ ν {ι} 6

25 2.2. A Group of Transformations on Copulas Proof. Assertions () an (2) immeiately follow from Lemma We now prove (3). It is obvious that ι Γ π Γ ν. Assume that there exists some γ Γ π Γ ν such that γ ι an consier the copula E(u) Π(u) + i u 2 i ( u i ) iscusse in Example Since γ Γ π we have γ(e) E, an since γ Γ ν an γ ι we have γ(e) ι(e) E. Thus, Γ π Γ ν {ι}. Lemma yiels Γ {ϕ Φ ϕ π ν for some π Γ π an some ν Γ ν } {ϕ Φ ϕ ν π for some π Γ π an some ν Γ ν } an the representation of a symmetry in terms of a permutation an a reflection is unique which is again a consequence of Lemma 2.2.7: Lemma. () For every γ Γ there exist unique π Γ π an ν Γ ν such that γ π ν. (2) For every γ Γ there exist unique π Γ π an ν Γ ν such that γ ν π. The reflection which can be expresse as the finite composition of partial reflections containing every partial reflection exactly once is calle the total reflection. We enote the total reflection by τ The total reflection is an involution an we have τ ν k k Moreover, we put Γ τ {ι, τ} Theorem. Γ is the smallest subgroup of Φ containing all transpositions an all partial reflections. Moreover, Γ! 2 an the center of Γ is Γ τ. Proof. Consier γ, γ Γ. As a consequence of Lemma 2.2.8, there exist unique π, π Γ π an ν, ν Γ ν such that γ π ν an γ π ν, an Lemma yiels the existence of some π Γ π an some ν Γ ν such that ν π π ν. We then obtain γ γ π ν π ν π π ν ν Since π π Γ π an ν ν Γ ν we have γ γ Γ, an thus, Lemma 2.. implies that Γ is a subsemigroup of Φ with neutral element ι. Further, let π Γ π be the inverse 7

26 Chapter 2. A Group of Transformations element of π an let ν Γ ν be the inverse element of ν. Set γ ν π. Then γ Γ an we obtain γ γ π ν ν π ι ν π π ν γ γ Thus, Γ is a subgroup of Φ an since Γ π Γ ν {ι} we have Γ Γ π Γ ν! 2. We now prove that the center Z(Γ) of Γ is Γ τ. First of all, Lemma implies π i,j τ π i,j ν k k ν k π i,j ν i ν j k k {i,j} ν k ν j ν i π i,j τ π i,j k k {i,j} for all i, j {,..., } with i j, an hence π τ τ π for all π Γ π. Since τ commutes with every reflection as well, we obtain Γ τ Z(Γ). Moreover, recall that, for 2, Γ π {ι, π,2 } an π,2 commutes neither with ν nor with ν 2 (Lemma 2.2.6), an, for 3, the center of Γ π is trivial (Theorem 2.2.2). In both cases, we hence obtain Z(Γ) Γ π {ι}, an thus, the subgroup Z(Γ) lies in the largest subgroup Λ Γ of Γ which satisfies Λ Γ π {ι}. We now prove that Γ ν is the largest subgroup satisfying this property. Recall that Γ ν Γ π {ι}, an assume that there exists a subgroup Λ Γ which satisfies Γ ν Λ an Λ Γ π {ι}. Then there exists some γ Λ satisfying γ π ν with unique π Γ π /{ι} an ν Γ ν. Let ν Λ be the inverse element of ν. Since Λ is a subgroup we obtain π π ν ν γ ν Λ which contraicts Λ Γ π {ι}. Thus, Z(Γ) Γ ν. Now, consier ν Γ ν /Γ τ. Then there exists some K {,..., } with K such that ν k K ν k. Choose i K an j {,..., }/K. Lemma then yiels Further, consier the copula π i,j ν π i,j ν k k K π i,j k K/{i} ν k ν i ν k π i,j ν i k K/{i} k K/{i} ν k ν j π i,j E(u) Π(u) + i u 2 i ( u i ) iscusse in Example Recall that E satisfies π(e) E for all π Γ π an ν(e) ν(e) for all ν, ν Γ ν with ν ν. We then obtain (π i,j ν)(e) ( ν k ν j π i,j )(E) k K/{i} ( ν k ν j )(E) k K/{i} ν(e) (ν π i,j )(E) This yiels π i,j ν ν π i,j, an thus, Z(Γ) Γ τ. 8

27 2.3. Subgroups an Generators The group Γ is a representation of the hyperoctaheral group with! 2 elements. For a etaile iscussion of the hyperoctaheral group incluing its structure an representations we refer to Young [930], Kerber [97] an Baake [984]. Remark. The construction of the group Γ can be generalize to transformations acting on the set of functions I R which are solely groune. In this case, Γ can be applie to semi copulas an quasi copulas. 2.3 Subgroups an Generators In Section 2.2 we constructe the group Γ an consiere first important subgroups. In this section we introuce further subgroups of Γ an focus on minimal systems of generators. This is of interest since it simplifies the verification that a given copula is invariant with respect to all transformations in a subgroup of Γ. We conclue this section with some results concerning convexity an convergence. The present section contains results an, in aition, several formulations of the paper Fuchs [204]. This inclues Lemma 2.3., Examples an Lemma In aition to the subgroups Γ π, Γ ν an Γ τ consiere before, we efine the set Γ π,τ {γ Γ γ π ϕ for some π Γ π an some ϕ Γ τ } which turns out to be of ecisive importance throughout this work. Since Γ τ is the center of Γ, we have Γ π,τ {γ Γ γ ϕ π for some π Γ π an some ϕ Γ τ } an this implies that Γ π,τ is a subgroup of Γ. Moreover, Γ π,τ 2!. As a counterpart to the total reflection τ, we enote by ψ π i,i+ i the total permutation. The total permutation is a permutation an satisfies We further consier the sets i ψ ι Γ ψ {γ Γ γ m ψ for some m {,..., }} i Γ ψ,ν {γ Γ γ ϕ ν for some ϕ Γ ψ an some ν Γ ν } Γ ψ,τ {γ Γ γ ϕ ϕ 2 for some ϕ Γ ψ an some ϕ 2 Γ τ } which are also subgroups of Γ: 2.3. Lemma. Γ ψ, Γ ψ,ν an Γ ψ,τ are subgroups of Γ. 9

28 Chapter 2. A Group of Transformations Proof. It follows from i ψ ι that Γ ψ is a subgroup of Γ. Since for all m {,..., }, an hence ν m i ψ ν m (m ) ψ i ν k l k ψ l k ψ ν l i for all k, l {,..., } such that k l, we also obtain that Γ ψ,ν is a subgroup of Γ. That Γ ψ,τ is a subgroup of Γ follows from the fact that Γ τ is the center of Γ. Moreover, Γ π,τ Γ ψ,ν Γ ψ,τ, Γ π Γ ψ,τ Γ ψ, Γ ψ,τ Γ ν Γ τ an Γ ψ Γ τ {ι}. The following figure isplays the lattice of the subgroups of Γ consiere before: Γ i Γ π,τ Γ ψ,ν Γ π Γ ψ,τ Γ ν Γ ψ Γ τ {ι} For each of these subgroups we give a minimal system of generators: Theorem. () The set {ψ} is a minimal generator of Γ ψ. (2) The set {τ} is a minimal generator of Γ τ. (3) For every i {,..., } the set {π i,i+, ψ} is a minimal generator of Γ π. (4) The set {ψ, τ} is a minimal generator of Γ ψ,τ. (5) The set {ν l, l {,..., }} is a minimal generator of Γ ν. (6) For every i {,..., } the set {π i,i+, ψ, τ} is a minimal generator of Γ π,τ. (7) For every k {,..., } the set {ψ, ν k } is a minimal generator of Γ ψ,ν. (8) For every i {,..., } an every k {,..., } the set {π i,i+, ψ, ν k } is a minimal generator of Γ. 20

29 2.3. Subgroups an Generators Proof. Assertions () an (2) are evient from the efinitions of Γ ψ an Γ τ. We now prove (3). To this en, consier i {,..., }. Since for all j {,..., }, an hence π,2 j l π j,j+ (j ) l ψ i ψ π j,j+ (j ) ψ l l ψ π i,i+ (i ) l ψ j ψ l for all j {,..., }, we obtain that {π i,i+, ψ} generates {π i,i+, i {,..., }}. Further, every transposition an hence every permutation can be expresse as a finite composition of transpositions {π i,i+, i {,..., }}. Thus, the set {π i,i+, ψ} also generates Γ π. That this generator is minimal follows from () an the fact that π i,i+ is an involution. This proves (3). Assertion (4) is an immeiate consequence of the efinition of Γ ψ,τ, () an (2). We now prove (5). Since every reflection can be expresse as a finite composition of partial reflections containing every partial reflection at most once, it is evient that {ν l, l {,..., }} generates Γ ν. Moreover, since τ Γ ν is the reflection containing every partial reflection exactly once, we obtain that the generator {ν l, l {,..., }} is minimal. This proves (5). Assertion (6) follows from (3), (2) an the efinition of Γ π,τ together with (3), (4) an the fact that π i,i+ an τ are involutions. Moreover, we prove (7). To this en, consier k {,..., }. Since ν l k l ψ ν k (k l) ψ i i for all l {,..., } such that l < k, as well as ν l (l k) i ψ ν k l k ψ for all l {,..., } such that k < l, we obtain that {ψ, ν k } generates {ν l, l {,..., }} an hence Γ ν as well as Γ ψ,ν. That this generator is minimal follows from () an the fact that ν k is an involution. This proves (7). Finally, consier i {,..., } an k {,..., }. Then {π i,i+, ψ, ν k } generates Γ π Γ ν an since every symmetry can be expresse as a composition of a permutation an a reflection, we hence obtain that {π i,i+, ψ, ν k } is a generator of Γ. That this generator is minimal follows from (3), (7) an the fact that π i,i+ an ν k are involutions. This proves (8). Assertions (3) an (8) of Theorem are essentially ue to Baake [984]. 2 i