A decomposition result for the Haar distribution on the orthogonal group

1 A Decomposition Result for the Haar Distribution o n the Orthogonal Group by Morris L. Eaton & Robb J. Muir head School of Statistics Statistical Res earch and Consulting Center University of Minnesota Pfizer Inc. Abstract Let Γ be a Haar distributed random matrix on the group p O of p p × real orthogonal matrice s. Partition Γ into four blocks, ( ) ( ) 1 1 : , 1 1 : , 1 1 : 21 12 11 × − Γ − × Γ × Γ p p and () () , 1 1 : 22 − × − Γ p p so . 22 21 12 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Γ Γ Γ Γ = Γ The marginal distribution of 11 Γ is well known. In this pape r, we give the conditional distribution of () 12 21 , Γ Γ giv en 11 Γ , and the conditional distribution of 22 Γ given () . , , 11 12 21 Γ Γ Γ This conditional specification uniquely determines the Haar distribution on p O . The two conditional distributions involve well known probability distributions – namely, the uniform distribution on the unit sphere { } 1 1 1 = ∈ = − − x R x p p S and the Haar distribution on 2 − p O . Our results show how to construct the Haar distribution on p O from the Haar distribution on 2 − p O coupled with the uniform distribution on . 1 − p S 1. Introduction and Summary The focus of this paper is the Haar p robability distribution on the group p O of p p × real orthogonal matrices. The use of this group and the Haar distribution in multivariate statistical analysis has a long history, with James (1954) and W ijsman (1957) being two important earl y contributions. A standard description of the Haa r distribution on p O is in terms of invariant differential forms – see Farrell ( 1985) for a s y stematic de velopment and excellent history of this approach in multivariate ana l y sis. A useful alternative is the use of random matrices, t he mu ltivariate normal distribution, and invariance properties of the objects under stud y . For example, see E a ton (1983) and Eaton (1989, Chapter 7). The 2 primary technical tools used in this pape r stem from the invariance considerations discussed at length in Eaton (1989). To describe the problem under conside ra tion in this paper, suppose the ran dom p p × orthogonal matrix Γ has the Haar distribution. This di stributi on is characteriz ed by its invariance. To be m ore precise, let ( ) ⋅ L denote the distribution (or probability law) of , " " ⋅ where " " ⋅ can be a r andom variable, a random vector , a random matrix, etc. Using the L -notation, the Haar probability distribution is c haracterize d by ( ) ( ) ( ) 2 1 g g Γ = Γ = Γ L L L for all . , 2 1 p g g O ∈ In other words, the Haar distribu tion is the unique invariant (ri ght or left) probability distribution on p O . In all that follows, we will assume that + ∈ Γ p O , where () . 1 , 1 , 11 22 21 12 11 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∈ = + h h h h h h h p p O O Note that + − p p O O is a set of Haar probability zero. In the arguments below, this set of probability zero has been removed from the s ample space of Γ. To describe the results in this paper, pa rtition + ∈ Γ p O as ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Γ Γ Γ Γ = Γ 22 21 12 11 where () ( ) 1 1 is , 1 1 is , 1 1 is 21 12 11 × − Γ − × Γ × Γ p p and ( ) ( ) . 1 1 is 22 − × − Γ p p The marginal distribution of 11 Γ is well known – see below following Theorem 1.1 . (It is well known even if 11 Γ is t s × with p t s ≤ + ; see Mitra (1970), Khatri (1970) and Eaton (1989, Chapter 7).) Thus, we will proce ed with ( ) 11 Γ L being specified. In what fol lows, the notation () ∗ ⋅ L is used for the conditional distribution of " " ⋅ given ". " ∗ The basic results in this paper provide a complete description of the two conditional distributions ( ) 11 12 21 , Γ Γ Γ L (1.1) and ( ) . 11 12 21 22 , , Γ Γ Γ Γ L (1.2) 3 A moment’s reflection will convince the reader th at knowing ( ) 11 Γ L , (1.1) and (1. 2) determines the Haar distribution on p O and conversel y. Here is a rigorous speci fication of (1.1). Let 1 U and 2 U be independent, identically distributed ( iid ) and uniform on the unit sphere { } 1 1 1 = ∈ = − − x R x p p S . Theorem 1.1 In the notation above and with ( ) 1 , 1 11 − ∈ Γ fixed, () ( ) ( ) ( ) 2 2 / 1 2 11 1 2 / 1 2 11 11 12 21 1 , 1 , U U ′ Γ − Γ − = Γ Γ Γ L L (1.3) where 2 U ′ is the transpose of . 2 U The above result asserts t hat ( ) 11 12 21 , , Γ Γ Γ L can be generated as follows: (i) First, draw 11 Γ from the densit y (see Eaton (1989, Proposition 7.3)) () ( ) () () () () () 1 1 1 2 / 3 2 2 1 2 1 2 1 < − − Γ Γ Γ = − x x p p p x f p , (ii) Next, draw 1 U and 2 U which are iid uniform on . 1 − p S Then use (1.3) to speci f y the conditional distribution of ( ) 12 21 , Γ Γ giv en . 11 Γ It is obvious that (i) and (ii) determine ( ) . 11 12 21 , , Γ Γ Γ L Our next task is to specify the conditional distribution (1.2). To this end, fix the values of . 12 21 11 , , Γ Γ Γ and recall that ( ) 1 , 1 11 − ∈ Γ . Le t 1 1 − ∈ p h O be an orthogonal transformation satisfying , 1 1 21 2 11 1 1 Γ Γ − = ε h (1.4) where . 0 0 1 1 1 − ∈ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = p R M ε Also, let 1 2 − ∈ p h O satisfy 4 . 1 1 12 2 11 1 2 Γ ′ Γ − = ε h (1.5) That 1 h and 2 h exist and depend onl y on the value of ( ) 12 21 , Γ Γ is demonstrated in Proposition A.2 in the Appendix. Finally, let the ( ) ( ) 2 2 − × − p p random matrix ∆ have the Haar distribution on . 2 − p O Theorem 1.2: In the notation above with 12 21 11 , , Γ Γ Γ fixed, a version of the conditional distribution of 22 Γ giv en () 11 12 21 , , Γ Γ Γ is the distribution ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ Γ − 2 11 1 0 0 h h L (1.6) where ∆ is Haar distributed on . 2 − p O Here 1 h and 2 h are given in (1.4) and (1.5) respectivel y. The proofs of both Theorem 1.1 and Theorem 1. 2 rel y to a large extent on some fairly well known notions from group theory and invari ant measures. In Section 2, we present the underl y ing group action that provides the appropriate setting for our p roofs. It is assumed that the reader is somewhat familia r with the standard notions of left group action, existence and uni queness of invariant m easures in the com pact case , and the basic representation result given in Theorem 4.4 in Eaton (1989, Chapter 4). The proof of Theorem 1.1, given in Section 3, involves little more than the standard assertion that the uniform probability distribution on 1 − p S is the unique orthogona lly invariant probability measure on . 1 − p S Our proof of Theorem 1.2 is somewhat more involved. I t depends on a gen eral constructive method for describin g an invari ant conditional distribution, given the value of an equivariant statistic. This method, whic h we believe is new, is pres ented rather abstractl y in the first portion of Section 3. A direct application of the method provides a proof of Theorem 1.2. Finally, we note here that we obtained have versions of Theorems 1.1 and 1.2 for the case where 11 Γ is 2 / 1 , p q q q < < × with . The results are av ailable from the authors. 2. A Group Acti on on p O We begin this section with a description of an inv ariance propert y of the conditional distribution () 11 Γ Γ L on + p O . As above, Γ has the H aar distribution on + p O and Γ has 5 been partitioned a s in Section 1. Le t H be the compact matrix group whose p p × elements h have the form . , 0 0 1 1 1 1 − ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = p h h h O Obviously H is a subgroup of p O so that ( ) ( ) k h ′ Γ = Γ L L (2.1) for all H. ∈ k h , The reason for the transpose on k in (2.1) is so that the action of the product group H H ⊗ on + p O given by k h ′ Γ → Γ (2.2) is in fact a left action. (See Eaton (1989, pp.19-20 ) for the distinction between left and right actions.) The action (2.2) can be expressed in terms of the blocks of Γ and the two () () 1 1 − × − p p lower right blocks of h and k : . 0 0 1 , 0 0 1 1 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = k k h h The action on the blocks is 1 22 1 22 1 12 12 21 1 21 11 11 k h k h ′ Γ → Γ ′ Γ → Γ Γ → Γ Γ → Γ (2.3) For each () , 1 , 1 − ∈ γ let γ X be the subset of + p O defined b y { } . 11 γ γ = Γ ∈ Γ = + p O X (2.4) It is clea r that H H ⊗ acts on γ X for each , γ with the action being given by (2.2), or equivalently, (2.3). Our first result implies that the ac tion of H H ⊗ on γ X is transitive. Proposition 2.1: Consider + ∈ p O ψ with 6 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 22 21 12 11 ψ ψ ψ ψ ψ partitioned as Γ is partitioned. Given ψ , define 0 ψ by () ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ′ − = ≡ * 22 1 2 11 1 2 11 11 0 11 0 1 1 ψ ε ψ ε ψ ψ ψ ψ ψ (2.5) where () () . 1 1 0 0 2 11 * 22 − × − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − p p I p ψ ψ Then there is an H H ⊗ ∈ ⊗ k h such that ( ) , 0 ψ ψ = ⊗ k h where () ψ k h ⊗ is specified b y (2.3). Proof: In the notation of (2.3), () . 1 22 1 21 1 1 12 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ′ = ⊗ k h h k k h ψ ψ ψ ψ ψ (2.6) First p ick 1 1 ~ h h = so that 1 2 11 21 1 1 ~ ε ψ ψ − = h and pick 1 1 ~ k k = so that . 1 ~ 1 2 11 1 12 ε ψ ψ ′ − = ′ k Then () . ~ ~ 1 1 ~ ~ ~ 1 22 1 1 2 11 1 2 11 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ − ′ − = ⊗ ≡ k h k h ψ ε ψ ε ψ ψ ψ ψ The fact that p O ∈ ψ ~ now implies that 1 22 1 22 ~ ~ ~ k h ′ = ψ ψ can be written 7 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ − = 22 11 22 ~ 0 0 ~ ψ ψ where . ~ 2 22 − ∈ ∆ p O Next pick H ∈ * h to be ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ ′ = 22 2 * ~ 0 0 I h and pick H. ∈ = p I k * Then a routine calcul ation shows that ( ) ( ) 0 * * ~ ~ ψ ψ = ⊗ ⊗ k h k h where 0 ψ is given in (2.5). This completes the proof. The following three propositions are eas y consequ ences of Proposition 2.1 coupled with standard invariance techniques. For some background material se e Eaton (1989) (Section 2.3 for a discussion of max imal invari ants and orbits, Section 4.1 for material on cross sections, and Theorem 4.1 for a r epresentation result). Proposition 2.2: Define f on + p O by ( ) ( ) . 11 0 + ∈ = p f O ψ ψ ψ Then f is a maximal invariant under the action of H. H ⊗ Proof: The invari ance o f f is obvious. Consider ψ and ξ in + p O and suppose ( ) ( ) ( ) , 11 0 ξ ψ ψ ψ f f = = so that . 11 11 ξ ψ = Use Proposition 2.1 to pick k h ⊗ and k h ~ ~ ⊗ so that () ( ) ( ) . ~ ~ 11 0 ξ ψ ψ ψ k h k h ⊗ = = ⊗ Then () () ξ ψ = ⊗ ⊗ − k h k h 1 ~ ~ so that f is a maximal invaria nt. This completes the proof. Proposition 2.3: The set ( ) ( ) { } 1 , 1 11 11 − ∈ ψ ψ ψ 0 8 is a measurable cross se ction of + p O . Proof: This is obvious from Propositions 2.1 and 2.2. It is now clear that t he orbi t of a point ψ in + p O is just , 11 ψ X and H H ⊗ acts transitively on ea ch γ X . Finally, let U have the un iform (Haar) distribution on the compact group H H ⊗ . Obviously, U has the same distribution as 2 1 V V ⊗ where 1 V and 2 V are iid Haar on H. Theorem 4.4 in Eaton (1989) immediatel y implies the following: Proposition 2.4: For fi xed ( ) , 1 , 1 11 − ∈ Γ ( ) ( ) 11 0 Γ ψ U L (2.7) serves as a version of the conditional distribution ( ) 11 Γ Γ L (2.8) when Γ is Haar on + p O . Note that the distribution (2.7) is on + p O but of course is concentrated on the or bit of () . 11 0 Γ ψ The distribution (2.8) can be in terpret ed as a distribution on , 11 Γ X or equivalently, as a distribution concentrated on th e orbit of ( ) . 11 0 Γ ψ In all that follows, we will take (2.7) to be a version of (2.8) and will treat (2.8) as a distribution on . 11 Γ X An immediate consequence of Pr oposition 2.4 is the following: Corollary 2.5: The cond itional distribution (2.8) on 11 Γ X is invariant under the action of H H ⊗ on ; 11 Γ X that is, ( ) ( ) 11 11 Γ ′ Γ = Γ Γ k h L L (2.9) for all H. ∈ k h , I n particular, by marg inalization, () ( ) 11 1 12 21 1 11 12 21 , , Γ ′ Γ Γ = Γ Γ Γ k h L L (2.10) for all H. ∈ k h , 9 The proof of Theorem 1.1 is now a routine ar gument using standard invariance techniques. Given , 11 Γ the vectors 21 Γ and 12 Γ satisfy . 1 2 11 2 12 2 21 Γ − = Γ = Γ Le t Z be this space of values for () . , 12 21 Γ Γ The distribution specified by (2.7) is invariant un der the action of the compact group H. H G ⊗ = To conclude uniqueness of this distribution and the conclusion of Theorem 1.1, note that Z is a topological left homogeneous space under this action. Here, we ar e using the term inology of Nachbin (1965, p. 128) which is explained in detail at the beginning of the nex t section. The reader should n ote that γ X is also a left homogeneous space unde r the action of G on . γ X This fact is used in the next section and in the Appendix . 3. An Inv ariant C onditi onal D ist ributi on We begin this section with a general result c oncerning the existence of invariant conditional distributions under ra ther natural invariance assumptions. This result is then used to provide a proof o f Theorem 1.2. Consider a Pol ish space () B X , which is acted upon topologicall y by a compact group G that is also a P olish space. H ere B is the σ -algebra of Borel sets of X. It i s assumed that X is a topological left homogeneous space – se e Nachbin (1965, p. 128) for a discussion of this terminology. In particular, G is assumed to be transitive on X and the map X G → : x T given by ( ) gx g T x = is assumed to be an open mapping. Under these assumptions, t here exi sts a unique G -invariant probability measure P defined on B . The notation () P X = L means that the random object X ∈ X has distribution P. Of course, () ( ) gX X L L = for all g since P is G -invariant. Next, we consider a continuous mapping t from X onto a Polish space ( ) . , C Y It is assumed that t is an equivariant map – that is, we assume () ( ) ( ) ( ) G. ∈ = ⇒ = g gx t gx t x t x t all for 2 1 2 1 (3.1) This assumption allows us to induce a group action on Y. The basic idea is the following. Given Y, ∈ y there is an X ∈ x such that ( ) y x t = since t is onto. Now, we simply define gy to be t ( gx ). It is assumption (3.1) that allows us to esta blish that this definition of gy , namel y ( ) ( ) , x gt gx t gy = = (3.2) is unambiguous. See Eaton (1989, Theorem 2.4 on page 32 and page 35 of Section 2.4) for details and some further discussion. In all that follows, we assume that Y is also a topological left homogeneous space und er the action of G . 10 Remark 3.1: As motivation for the above assumption, we note that the main application of the material to follow is for the case when X is γ X given in Section 2. The mapping t is given by ( ) ( ) 12 21 , , Γ Γ = Γ γ t for . γ X ∈ Γ Of course, H H G ⊗ = in this application. The main theoretica l result of this section establishes the e xistence of an invariant version of the conditional distribution of X given ( ) Y. ∈ = y X t More precisely, let () () y X t X = L denote some version of the conditional distribution when X has distribution P above. In what follows, we will show that there is a Marko v kernel, ( ) y R ⋅ on , Y B × that serves as a versi on of ( ) ( ) y X t X = L and is invariant in the se nse that ( ) ( ) gy gB R y B R = (3.3) for all Borel sets , B ∈ B for all Y ∈ y and for . G ∈ g It is this invaria nce, when applied to the situation of Section 1, that underlies the proof of Theor em 1.2. We now proceed wit h some technical det ails. Proposition 3.1: The action of the group G on Y is transitive. Proof: Consider 1 y and 2 y in Y We need to show that there is a G ∈ g so that . 2 1 y gy = Because t is an onto m ap, there ex ist 1 x and 2 x in X so that () i i y x t = for i=1,2. But G is transitive on X by assumption, so 2 1 x gx = for some g . Using (3.2), we have ( ) ( ) ( ) 1 1 1 2 2 gy x gt gx t x t y = = = = and the proof is complete. Proposition 3.2: Let () () . X t Q L = Then Q is an invariant probability measure on ( ) . C Y, Proof: For C ∈ C and G, ∈ g () ( ) {} ( ) () ( ) { } ( ) { } ( ) . 1 1 C g Q C g X t P C X gt P C gX t P C X t P C Q − − = ∈ = ∈ = ∈ = ∈ = 11 Proposition 3.3: For Y, ∈ y let ( ) { } . y x t x y = = X Then . y gy g X X = (3.4) Proof: Using the equivariance o f t, we have () {} () { } ( ) ( ) { } () { } . 1 1 1 y gy g y u t gu y x g t x g g y x t g x gy x t x X X = = = = = = = = = − − − This completes the proof. Now, we proceed with the descri ption of the tr ansition function ( ) y R ⋅ that is to serve as a version of () () . y X t X = L (i) Fix Y ∈ 0 y and let X ∈ 0 x be an y point such that ( ) . 0 0 y x t = (ii) Let G G ⊆ 0 be the group { } . G 0 0 0 y gy g = = Clearl y 0 G is a compact subgroup of G. Let 0 U be the unique random element of 0 G with the Haar (on 0 G ) probability distribution. (iii) From Proposition A.1 in the Appe ndix, there is a measurable m ap k from Y into G such that () y y y k = 0 for all Y. ∈ y (iv) For Y, ∈ y consider the random variable ( ) 0 0 x U y k Z y = (3.5) and let () y R ⋅ denote the distribution of X. ∈ y Z In other words, for B, ∈ B () ( ) { } ( ) ( ) { } , Pr Pr 1 0 0 0 0 B y k x U B x U y k y B R − ∈ = ∈ = where “Pr” r efers to the Haar distribution of 0 U on 0 G . The measurability of () ⋅ k insures that () ⋅ ⋅ R is a Markov kernel. The following result establishes some basic properties of ( ) ⋅ ⋅ R . Proposition 3.4: The Markov kern el R satisfies the following: (i) () ( ) y B R gy gB R = (3.6) for all B , g and y . 12 (ii) ( ) 1 = y R y X (3.7) for all y . Proof: To establish (3.6), consider fix ed g and y . Then () () {} ( ) { } . Pr Pr 1 0 0 1 0 0 B g x U gy k g B x U gy k gy B R − − ∈ = ∈ = But () y k satisfies () y y y k = 0 and ( ) . 0 1 y y gy k g = − Therefore ( )( ) 0 1 g y k gy k g = − for some . 0 0 G ∈ g Thus, () ( ) { } . Pr 1 0 0 0 B g x U g y k gy B R − ∈ = Since () ( ) , . 0 0 0 U U g L L = we conclude that () ( ) { } ( ) . Pr 1 1 0 0 y B g R B g x U y k gy B R − − = ∈ = Thus (3.6) holds. For (3.7), first observ e that (3.7) holds when . 0 y y = Since , gy y g X X = (3.6) yields ( ) ( ) ( ) . 1 0 0 0 0 0 0 gy R gy g R y R gy y y X X X = = = The transitivity of G on Y now gives (3.7). Note that (3.6) implies () ( ) ( ) ( ) y g dx R x f y dx R x g f 1 1 − − ∫ ∫ = (3.8) for all G ∈ g and for all bounded measurable f. The vali dity of (3.8) follows from (3.6) and the standard approx imati on of bounded measurable functions b y linear combinations of indicator functions. Theorem 3.1: The Mark ov kernel R serves as a re gular version of the cond itional distribution of X given () . y X t = Remark: We are usin g the terminology “regular conditional distribution” in the sense defined in Section 8, Chapte r 5 of Parthasarath y (1967). 13 Proof: First recall that P is the unique G -invariant probability distribution on X. Let () X K denote all the bounded m easurable functions on X and define the following integral on () X K : For () , X K f ∈ let () ( ) ( ) ( ) . dy Q y dx R x f f J ∫∫ = Y X (3.9) The group G acts on () X K via the action ( ) ( ) ( ) . 1 x g f x gf − = Using the invarian ce of Q and (3.8), it is easy to show that ( ) ( ) gf J f J = for all G ∈ g and () . X K f ∈ Thus J given by (3.9) is an invariant probability integra l on () . X K By uniqueness, we have ( ) ( ) ( ) f J dx P x f = ∫ (3.10) for all () . X K f ∈ Now, (3.10) coupled with (3.7 ) and Theo rem 8.1 on page 147 of Parthasarath y (1967) show that ( ) y R ⋅ is a regular conditional distribution and is unique up to sets of Q -measure zero. Thi s completes the proof. Example 3.1 (The proof of Theorem 1.2): Here, the general situ ation considered above is specializ ed to the case consi dered in Theorem 1.2. The basic ide a is to identi fy the random variable “ y Z ” in (3.5), since it is the distribution of y Z that provides the conditional distribution of X giv en ( ) . y X t = To this end, we need a careful specific ation of the spaces involved and a clea r description of the basic objects “ , , , , 0 0 0 0 U x y G and () y k ”, all of which go into the de finition of y Z and its distribution. To begin, we again let Γ have the Haar distribution on + p O and fix the value of 11 Γ to be () 1 , 1 − ∈ γ . From the results in Section 2, the conditional distribution of Γ given γ = Γ 11 is concentrat ed on the comp act set γ X (see (2.4)) and is the unique invariant distribution under the transitive action of the com pact group H H G ⊗ = on γ X . For this exampl e, the set “ X ” is γ X and is easily shown to be a left homogeneous spac e under the action of H. H ⊗ Next, for , 22 21 12 γ γ X ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Γ Γ Γ = Γ 14 define the map t by ( ) { } Y. ∈ Γ Γ = Γ 12 21 , , γ t (3.11) Here, Y is the comp act space {} () ( ) , 2 1 S S × × γ where () { } 2 2 1 1 1 , γ − = ∈ = − u R u u p S (3.12) and () { } . 1 , 2 2 1 2 γ − = ∈ ′ = − u R u u p S (3.13) Elements of () 1 S are column vectors while el ements of () 2 S are row vect ors. That t in (3.11) is an equivariant map is readil y verified, and the action of G on Y is of course {} { } k h ′ Γ Γ → Γ Γ 12 21 12 21 , , , , γ γ (3.14) for G. H H = ⊗ = ⊗ k h The conditional distribution of Γ given ( ) Γ t is what is desired. Now, we follow the procedure that leads to “ y Z ”. First, let ( ) 1 2 1 2 0 1 , 1 , ε γ ε γ γ ′ − − = y (3.15) and let ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − = − 0 0 0 0 0 0 1 0 0 1 2 2 2 0 p I x M M L L γ γ γ γ (3.16) It is an eas y argument to show that { } 0 0 0 y gy g = = G is just , H H 0 0 ⊗ where H H ⊆ 0 is the subgroup . , 0 0 2 22 22 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∈ = − p p h h I h h O O H (3.17) The random gr oup element 0 0 G ∈ U with the Haar distribution is just 15 , 0 0 2 . 0 1 , 0 0 H H ⊗ ∈ ⊗ = V V U (3.18) where 1 , 0 V and 2 , 0 V are iid Haar on . H 0 It is obvious that for i = 1,2 () ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ = 0 0 2 , 0 I V i L L (3.19) where ∆ is Haar on . 2 − p O We now see that with 0 x given by (3.16) and 0 U given by (3.18), () ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∆ − − − = 0 0 0 0 0 0 1 0 0 1 2 2 0 0 M M L L γ γ γ γ L L x U (3.20) where ∆ is Haar on . 2 − p O The next step in this proof of Theorem 1. 2 is the calculation of a Borel measura ble function () G ∈ y k that satisfies () Y ∈ = y y y y k , 0 (3.21) for this example. With {} Y, ∈ Γ Γ = 12 21 , , γ y a direct application of t he material in Section A.2 shows that there is a () ( ) 1 1 − × − p p orthogonal matrix 1 h that satisfies ( ) 21 1 2 1 1 Γ = − ε γ h (3.22) and 1 h is a Borel function of . 21 Γ Similarly, ther e is a ( ) ( ) 1 1 − × − p p orthogonal matrix 2 h that satisfies ( ) 12 1 2 2 1 Γ ′ = − ε γ h (3.23) and 2 h is a Borel function of . 12 Γ Then, setting () , 0 0 1 0 0 1 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⊗ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = h h y k (3.24) 16 we see that (3.21) holds and () ⋅ k is a Borel function of Y. ∈ y With () y k given by (3.24), the random va riable y Z specified in (3.5) i s ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ Γ Γ = 2 22 1 21 12 h h Z y ψ γ (3.25) where () () 1 1 : 0 0 22 − × − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ − = p p γ ψ and ∆ is Haar on . 2 − p O By Theorem 3.1, the distribution of y Z serves as a version of the conditional distribution of Γ given ( ) . Γ t This completes the proof of Theorem 1.2. The above proof leads di rectly to an al gorithm for generating a Haa r distributed matrix Γ on , p O given a Haar distributed matrix ∆ on . 2 − p O Here is the algorithm: 1. Draw 11 Γ from the density ( ) p x f given in Section 1. 2. Next draw 1 U and 2 U iid uniform on 1 − p S and let . 1 , 1 2 2 11 12 1 2 11 21 U U ′ Γ − = Γ Γ − = Γ 3. Then construct the mat rices 1 h and 2 h as in (3.22) and (3.23) (b y applying Proposition A.2). Then the matrix , 2 * 22 1 21 12 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ Γ Γ Γ Γ = Γ h h (3.26) where , 0 0 11 * 22 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ Γ − = Γ (3.27) 17 is Haar distributed on . p O Appendix This appendix contains two technical results, the first of which establishes the existence of a measurable sel ector needed in the constructio n of the Markov kernel R in Section 3. Section A.1: In this section we consider a compact Hausdorf f group G and a topological left homogeneous Polish space Y under the left action of G on Y . Fix a point Y ∈ 0 y and let { } 0 0 0 y gy g = = G . (A.1) Then 0 G is obviously a compact subgroup of G . Fo r eac h Y, ∈ y let { } . 0 y gy g y = = G (A.2) Since y G is closed, it is a compact subset of G . Proposition A.1: There exists a Borel measurable m ap k fr om Y into G such that () y y k G ∈ for all Y. ∈ y Thus ( ) y y y k = 0 for all Y. ∈ y Proof: We will use the Kuratowski-R y ll-Naadzewski Theorem a s stated in Aliprantis and Border (1999, p.567, Theorem 17.13). Here is a sketch of the argument . Consider the correspondence c d efined on Y whose values are subsets of G , given by () y y c G = , with y G defined b y (A.2). (S ee Aliprantis and Border (1999, Chapter 16) for a discussion of correspondence.) Th en () y c is a closed correspondence sinc e each y G is a closed set. Recall tha t c is called weakly measurable (see Aliprantis and Bo rder (1999, p.558 and p.525)) if ( ) { } φ ≠ ∩ = U y c y V (A.3) is a Borel subset of Y for each open subset U of G . But, by assumption, the ma p 0 y T from G to Y defined b y () 0 0 y g g T y = is an open mapping. Henc e () U T y 0 is an open subset of Y and so is Borel. 18 Now, it is a routine argument to show ( ) U T V y 0 = where V is given b y (A.3), so V is open and hence Borel. Because G is transitive on Y , ( ) y c is not empty. Theorem 17.13 in Aliprantis and Border (1999) immediatel y implies the existence of a measurable selector k. (In other words, k i s a Borel measur able map from Y to G with () ( ) y c y k ∈ for each y .) This completes the proof. Section A.2: Here we give an explicit formula for a sy mmetric orthogonal matrix that interchanges two given v ectors of the same length. Consider vec tors u and v in p R with . 0 > = v u For , v u ≠ set v u w − = and define the p p × matrix h by ⎪ ⎩ ⎪ ⎨ ⎧ = ≠ ′ ′ − = v u I v u w w w w I h p p if if 2 (A.4) Proposition A.2: The matrix h is symmetric, orthog onal, and satisfies v hu = and . u hv = Proof: S ymmetry and orthog onalit y are obvious. The case of v u = is obvious, so assume . v u ≠ Since , w hw − = ( ) . v u v u h + − = − But ( ) v u v u h + = + because () . 0 = + ′ v u w Adding these two relations yields v hu = , so u hv = by symmetry and orthogonality. This completes the proof. Fix an vector p R v ∈ with 0 > v and let { } . v u R u p v = ∈ = S Then define * h on v S to p O by ( ) , * h u h = where h is given b y (A.4). I t is not difficult to show that the mapping * h is Borel measurable. References Aliprantis, C. D. and Border, K. C. (1999). Infinite Dimensional Analysis. Second edition. Springer-Verla g, New York. Eaton, M. L . (1983). Multivaraiate Statistics: A Vector Space Approach. 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