Wishart α-determinant, α-hafnian

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1 Wishart α-determinant, α-hafnian (, JST CREST) (, JST CREST), Wishart,. ( )Wishart,. determinant Hafnian analogue., ( )Wishart,. 1 Introduction, Wishart. p ν M = (µ 1,..., µ ν ) = (µ ij ) i=1,...,p p p Σ = (σ i,j ) fix = M tm. j=1,...,ν X 1 = (x i1 ) 1 i p, X 2 = (x i2 ) 1 i p,..., X ν = (x iν ) 1 i p ν p, N p (µ 1, Σ),..., N p (µ ν, Σ),, N p (µ i, Σ) µ i, Σ. X (X 1,..., X p ) ν p, W = (w ij ) W = X tx., W, W Wishart W p (ν, Σ, )., mean square matrix, = 0 Wishart W p (ν, Σ).,, W = X t X Wishart CWp (ν, Σ, ).,. x = (x 1,..., x m ) D. n = (n 1,..., n m ), x n = x n 1 1 xn m m E[x n 1 1 xn m m ] D (n- ). D, D., Wishart, Wishart : E[e tr(θw ) ] = det(i 2ΘΣ) ν 2 e 1 2 tr(i (I 2ΘΣ) 1 )Ω,, Θ p p symmetric parameter matrix, Ω ΣΩ = 1

2 . Wishart ( [2, 11] ). Wishart E[w i1,i 2 ]., determinant analogue. E[w i1,i 2 ]. [5]. 2 Notation of graphs,.,. v w, v w {v, w}. v w, {v, w} = {w, v}., self loop {v, v}. V, U, K V K V,U : K V,U = { {v, u} v V, u U, v u }, K V = K V,V = { {v, u} v u V }. V E K V G = (V, E ), vertex(e ) = { v V {v, u} E for some u V }. (V, K ). E K (V, K ) : {v, u}, {v, u } E = u = u., ( ) 1. M (V, K ) (V, K )., (V, K V ) M (V, K V ) M (V )., (V, K ) E vertex(e ) = V, E perfect. ( 1.) P (V, K ) (V, K ) perfect matchings. P (V ) = P(V, K V ).. v w (v, u). v u, (v, u) (u, v).,, self loop (v, v)., self loop. V, U, K V K V,U K V,U = { (v, u) v V, u U }, K V = K V,V = { (v, u) v, u V }. V E K V G = (V, E), start(e) end(e) 2

3 : start(e) = { v V (v, u) E for some u V }, end(e) = { u V (v, u) E for some v V }. (V, K). 2 E K (V, K) : (v, u), (v, u ) E = u = u (v, u), (v, u) E = v = v., ( ) ( ) 1. (V, K) M(V, K). E M(V, K) start(e) = V end(e) = V, perfect, P(V, K) (V, K) perfect matching. Perfect matching 1., M(V ) = M(V, K V ), P(V ) = P(V, K V ). Remark 2.1. V Z, V = { v v V }, V = { v v V }, l = 2l 1, l = 2l., (V, E) 2 ( V, V, { { v, ü} (v, u) E })., M(V, K) 2 ( V, V, { { v, ü} (v, u) E }). M(V, K). 3 Definition of our polynomials l Z, l = 2l 1, l = 2l. n Z>0 fix, V, V fix : V = [n] = { 1,..., n }, V = [n] = { 1,..., ṅ }, V = [n] { = 1,..., n }, V = V V = [n] [n] = [2n].,. E M(V ). (V, E). len(e) (V, E). V \ start(e) (V, E), V \ end(e)., Ě : Ě = { (v, u) K V \start(e),v \end(e) E u v. } K V. Remark 3.1. E M(V ), Ě : Ě M(V ), Ě E =, Ě E P(E), (V, E) (V, Ě E). 3

4 Remark 3.2. E M(V ), len(e) : len(e) = ((V, E) ) Ě., E P(V ), (i, j) E σ E (i) = j n S n σ E, E, len(e) σ E. (V, E) x = (x i,j ), weight monomial x E x E = (v,u) E. : Definition 3.3. K K V, det α (x, y; K) det α (x; K) : det α (x, y; K) = α n len(e) x E yě, det α (x; K) = E M(V,K) E P(V,K) x v,u α n len(e) x E., det α (x, y) = det α (x, y; K V ), det α (x) = det α (x; K V ). Remark 3.4. det α (x; K) = det α (x, 0; K). {,. { 1, 1},..., {ṅ, n} } K V, E V 0. E M (V ), Ě len(e ) : { } Ě = {v, u} K V \vertex(e ) E E 0 v u. len(e ) = ((V, E E 0) ) Ě. Remark 3.5. E M (V ), Ě : Ě M (V ), Ě E =, Ě E P (E ), (V, E E 0) (V, Ě E E 0). Remark 3.6. E M (V ), (V, E E 0)., (V, E E 0). E V \ vertex(e ) (V, E E 0). (V, E E 0) Ě. 4

5 (V, E ) x = (x i,j ), weight monomial x E x E = {v,u} E x v,u. v, u V x v,u = x u,v, x E well-defined. Definition 3.7. K K V, Hf α(x, y; K ) Hf α (x; K ) : Hf α (x, y; K ) = α n len(e ) x E, yě Hf α (x; K ) = E M (V,K ) E P (V,K ) α n len(e ) x E. Hf α (x, y) = Hf α (x, y; K V ), Hf α(x) = Hf α (x; K V ). Remark 3.8., Hf α (x; K ) = Hf α (x, 0; K ). Remark 3.9. A = (a ij ), α-determinant (or α-permanent) : det α (A) = σ S n α n len(σ) a 1,σ(1) a 2,σ(2) a n,σ(n). determinant permanent α-analogue ;, α-determinant α = 1 determinant, α = 1 permanent. (See also [13, 14].) Remark 3.2, α-determinant det α (A). [9] α-pfaffian, skew-symmetric matrix A : Pf α (A) = E P (V ) ( α) n len(e ) sgn(e )A E. sgn(e )A E E = { {x 1, x 1 },..., {x ṅ, x n } } x S 2n sgn(x)a x 1,x 1 a xṅ,x n, A skew symmetric sgn(e )A E x S 2n. α-pfaffian Pfaffian α-analogue,, α = 1 α-pfaffian Pfaffian Pf(A), i.e., sgn(x)a x 1 x 1 a xṅx n,., symmetric matrix B Hf α (B) Hf α (B) = α n len(e ) B E E P (V ),, α = 1, Hafnian Hf(B) = b x 1 x 1 b xṅx n. Hafnian analogue. 5

6 4 Main results det α (x, y), Hf α (x, y) Wishart. Propsition 4.1. W = (w i,j ) W p (ν, Σ, ),, W Wishart W p (ν, Σ, ). A B : a u,v = σ u,v, b u,v = δ u,v. : E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = ν n Hf ν 1(A, B) = Hf ν 1(νA, νb). : Theorem 4.2. A B : a u,v = σ iu,i v, b u,v = δ iu,i v. W W p (ν, Σ, ) : E[w i1,i 2 ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = ν n Hf ν 1(A, B) = Hf ν 1(νA, νb)., Wishart : Propsition 4.3. W = (w i,j ) CW p (ν, Σ, ), A, B : a u,v = σ u, v, b u,v = δ u, v. : E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = ν n det ν 1(A, B) = det ν 1(νA, νb)., : Theorem 4.4. A, B : a u,v = σ i u,i v, b u,v = δ i u,i v., W = (w i,j ) CW p (ν, Σ, ) : E[w i1,i 2 ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = ν n det ν 1(A, B) = det ν 1(νA, νb). Remark 4.5.,, Wishart,. [5, 12]. Remark 4.6. Wishart,, Wishart, Lu, Richards [7]; Graczyk, Letac, Massam [3, 4]; Vere-Jones [13]., Letac, Massam [6] Wishart,., Wishart,, Matsumoto [10]. 6

7 [1] Bai, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices, A review. Statist. Sinica, 9, [2] Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (An introduction). Ann. Math. Statist., 34, [3] Graczyk, P., Letac, G. and Massam, H. (2003). The complex Wishart distribution and the symmetric groups. Ann. Statist., 31, [4] Graczyk, P., Letac, G. and Massam, H. (2005). The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution. J. Theor. Probab., 18, [5] Kuriki, S. and Numata, Y. (2009). Graph representations for moments of noncentral Wishart distributions and their applications. Submitted. arxiv: v1 [6] Letac, G. and Massam, H. (2008). The noncentral Wishart as an exponential family, and its moments. J. Multivariate Anal., 99, [7] Lu, I-L. and Richards, D. St. P. (2001). MacMahon s master theorem, representation theory, and moments of Wishart distributions. Adv. Appl. Math., 27, [8] Maiwald, D. and Kraus, D. (2000). Calculation of moments of complex Wishart and complex inverse Wishart distributed matrices. IEE Proc.-Radar, Sonar Navig, 147, [9] Matsumoto, S. (2005) α-pfaffian, pfaffian point process and shifted Schur measure. Linear Algebra and its Applications, 403, [10] Matsumoto, S. (2010) General moments of the inverse real Wishart distribution and orthogonal Weingarten functions, arxiv: v2 [11] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. John Wiley & Sons [12] Numata, Y. and Kuriki, S. (2009). On formulas for moments of the Wishart distributions as weighted generating functions of matchings, DMTCS Proceedings, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), pp (online journal) [13] Vere-Jones, D. (1988). A generalization of permanents and determinants. Linear Algebra Appl., 111, [14] Vere-Jones, D. (1997). Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math., 26,

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