On formulas for moments of the Wishart distributions as weighted generating functions of matchings

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1 FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, On formuas for moments of the Wishart distributions as weighted generating functions of matchings Yasuhide NUMATA 1,3 and Satoshi KURIKI 2,3 1 Department of Mathematica Informatics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan. 2 The Institute of Statistica Mathematics, 10-3 Midoricho, Tachikawa, Tokyo , Japan. 3 Japan Science and Technoogy Agency (JST), CREST. Abstract. We consider the rea and compex noncentra Wishart distributions. The moments of these distributions are shown to be expressed as weighted generating functions of graphs associated with the Wishart distributions. We give some bijections between sets of graphs reated to moments of the rea Wishart distribution and the compex noncentra Wishart distribution. By means of the bijections, we see that cacuating these moments of a certain cass the rea Wishart distribution bois down to cacuations for the case of compex Wishart distributions. Résumé. Nous considérons es ois Wishart non-centrae rée et compexe. Les moments sont décrits comme fonctions génératrices de graphes associées avex es ois Wishart. Nous donnons bijections entre ensembes de graphes reatifs aux moments des ois Wishart non-centrae rée et compexe. Au moyen de a bijections, nous voyons que e cacu des moments d une certaine casse a oi Wishart rée deviennent e cacu de moments de oi Wishart compexes. Keywords: generating funtion; Hafnian; matching; moments formua; Wishart distribution. 1 Introduction First we reca the Wishart distributions which originate from the paper by Wishart [18]. Let X 1 = (x i1 ) 1 i p, X 2 = (x i2 ) 1 i p,..., X ν = (x iν ) 1 i p be p-dimensiona random coumn vectors distributed independenty according to the norma (Gauss) distribution N p (µ 1, Σ),..., N p (µ ν, Σ) with mean vectors µ 1 = (µ i1 ) 1 i p,..., µ ν = (µ iν ) 1 i p (respectivey) and a common covariance matrix Σ = (σ ij ). The distribution of a p p symmetric random matrix W = (w ij ) 1 i,j p defined by w ij = ν t=1 x itx jt is the rea noncentra Wishart distribution W p (ν, Σ, ), where = (δ ij ) 1 ij p is the mean square matrix defined by δ ij = ν t=1 µ itµ jt. The Wishart distribution for = 0 is said to be centra and is denoted by W p (ν, Σ). The matrix Ω = Σ 1 is caed the noncentraity matrix. It is usuay used instead of to parameterize the Wishart distribution. However, in this paper, we use (ν, Σ, ) for simpicity in describing formuas. The compex Wishart distribution CW p (ν, Σ, ) is defined as the distribution of some p p Hermitian random matrix constructed from random vectors distributed independenty according to the compex norma (compex Gauss) distributions c 2010 Discrete Mathematics and Theoretica Computer Science (DMTCS), Nancy, France

2 822 Yasuhide NUMATA and Satoshi KURIKI The moment generating function of the rea Wishart distribution is given by E[e tr(θw ) ] = det(i 2ΘΣ) ν 2 e 1 2 tr(i 2ΘΣ) 1 Θ, where Θ is a p p symmetric parameter matrix [13]. Simiary, the moment generating function of the compex Wishart distribution is given as foows: E[e tr(θw ) ] = det(i ΘΣ) ν e tr(i ΘΣ) 1 Θ, where Θ is a p p Hermitian parameter matrix. See aso [2] for the centra case. Our first objective is to describe the moments E[w i1,i 2 w i3,i 4 w i2n 1,i 2n ] of the Wishart distributions of genera degrees in expicit forms. Since the Wishart distribution is one of the most important distributions, it has been studied by many researchers not ony in the fied of mathematica statistics but aso in other fieds (e.g., [1, 11]). Its moments have been we studied, and methods to cacuate the moments in the centra cases have been deveoped by Lu and Richards [10]; Graczyk, Letac, and Massam [3, 4]; Vere-Jones [16]; and many other authors. In particuar, Graczyk, Letac and Massam [3, 4] deveoped a formua for the moments using the representation theory of symmetric group. More recenty, Letac and Massam [9] introduced a method to cacuate the moments of the noncentra Wishart distributions. In this paper, we introduce another formua for the moments of Wishart distribution; in our formua, the moments are described as specia vaues of the weighted generating function of matchings of graphs. Cacuation of the moments bois down to enumeration of graphs via our formuas. As an appication of our formuas, we construct some correspondences between some sets of graphs, which impies severa identities of moments. The organization of this paper is as foows. In Section 2, we introduce some notations for the graphs. In Section 3, we define the generating functions of matchings and give the main formuas, which are an extension of Takemura [15] deaing with the centra case. In Section 4.1, we give some correspondences between directed and undirected graphs, which impies equations between the moments of the compex Wishart distribution and the moments of the rea Wishart distribution for some specia parameter. In Sections 4.2 and 4.3, we consider the Wishart distribution with some degenerated parameters. We see that the cacuation of its moments is reduced to enumerating graphs satisfying some conditions. A part of this paper is taken from our previous paper [8]. Pease see the paper for the omitted proofs. 2 Notation of graphs In this paper, we consider both undirected and directed graphs. For Z, we define and by = 2 1 and = 2. Let us fix n Z >0. We aso fix sets V and V as foows: V = [n] = 1,..., n }, V = [n] = 1,..., ṅ }, V = [n] = 1,..., n }, and V = V V = [n] [n] = [2n]. We use V and V as the sets of vertices of directed and undirected graphs, respectivey. First we consider undirected graphs. For v w, the undirected edge between v and w is denoted by v, w} = w, v}. We do not consider undirected sef oops, i.e., v, v}. For sets W and U of vertices, we define sets K U and K W,U of undirected edges by K W,U = w, u} w W, u U, w u }, K U = K U,U = v, u} v u V }. We ca a pair G = (V, E ) of a finite set V and a subset E K V an undirected graph. For an undirected graph G = (V, E ), we define vertex(e ) by vertex(e ) = v V v, u} E for some u V }. Let (V, K ) be an undirected graph. We ca a subset E K a matching in (V, K ) if no two edges in E share a common vertex. We define M (V, K ) to be the set of matchings in (V, K ) and M (V ) to be the set M (V, K V ) of matchings

3 Formuas moments of Wishart distributions 823 in the compete graph (V, K V ). A matching E in (V, K ) is said to be perfect if vertex(e ) = V. We define P (V, K ) to be the set of perfect matchings in (V, K ) and P (V ) by P (V ) = P (V, K V ). Next we consider directed graphs. A directed edge from v to w is denoted by (v, u). For v u, (v, u) (u, v). In the case of directed graphs, we aso consider a directed sef oop (v, v). For a directed edge e = (v, u), we respectivey ca v and u a starting and end points of e. For sets W and U of vertices, we define sets K U and K W,U of directed edges by K W,U = (v, u) v W, u U }, K U = K U,U = (v, u) v, u U }. We ca a pair G = (V, E) of a finite set V and a subset E K V a directed graph. For a directed graph G = (V, E), we define start(e) and end(e) by start(e) = v V (v, u) E for some u V }, end(e) = u V (v, u) E for some v V }. For a directed graph (V, K), we ca a subset E K a matching in (V, K) if v v and u u for any two distinct directed edges (v, u) and (v, u ) E. We define M(V, K) to be the set of matchings in (V, K) and M(V ) by M(V ) = M(V, K V ). A matching E in (V, E) is said to be perfect if start(e) = V and end(e) = V. We define P(V, K) to be the set of perfect matchings in (V, K) and P(V ) by P(V ) = P(V, K V ). Remark 2.1 We can identify a directed graph (V, E) with a bipartite graph ( V, V, v, ü} (v, u) E }). i.e., a graph whose edges connect a vertex in V to a vertex in V. Via this identification, an eement in M(V, K) is identified with a matching in the bipartite graph. In this sense, we ca an eement in M(V, K) a matching in (V, K). 3 Weighted generating functions and moments First we consider undirected graphs to describe the moments of rea Wishart distributions. For an undirected graph (V, E ) and variabes x = (x i,j ), we define the weight monomia x E by x E = v,u} E x v,u. If x v,u = x u,v for v, u V, then the weight monomia x E is we-defined. We define E 0 to be 1, 1},..., ṅ, n} } K V, V. For E M (V ), we define Ě and en(e ) by Ě = } v, u} K V \vertex(e ) There exists a chain between v and u in E E 0. K V en(e ) = (the number of connected components in (V, E E 0)) Ě. Remark 3.1 For E M (V ), Ě can be defined as a subset of K V satisfying the foowing conditions: Ě M (V ), Ě E =, Ě E P (E ), The number of connected components in (V, E E 0) equas the number of connected components in (V, Ě E E 0).

4 824 Yasuhide NUMATA and Satoshi KURIKI Remark 3.2 For E M (V ), et us consider the undirected graph (V, E E 0) with mutipe edges. The connected components of (V, E E 0) are chains and cyces without chords. The number of cyces in (V, E E 0) equas en(e ). The vertices V \ vertex(e ) which do not appear in E are terminas of chains in (V, E E 0). The set of pairs of terminas of chains in (V, E E 0) equas Ě. Definition 3.3 For a set K K V of undirected edges, we define poynomias Φ K and Ψ K by Φ K (t, x, y) = t en(e ) x E yě, Ψ K (t, x) = t en(e ) x E. E M (V,K ) E P (V,K ) We aso respectivey define Φ (t, x, y) and Ψ (t, x) to be Φ K (t, x, y) and Ψ V K (t, x). V Remark 3.4 By definition, Ψ K (t, x) = Φ K (t, x, 0) for each K K V. We have the foowing formua that describes the moments of the rea noncentra Wishart distribution as the specia vaues of the weighted generating function. Theorem 3.5 Let W = (w i,j ) W p (ν, Σ, ), namey, et W be a random matrix distributed according to the rea noncentra Wishart distribution W p (ν, Σ, ). Then E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = Φ (t, x, y) = Φ (ν, Σ, ). t=ν, xu,v=σ u,v, y u,v=δ u,v Coroary 3.6 For W W p (ν, Σ, ) E[w i1,i 2 w i3,i 4 w i2n 1,i 2n ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = Φ (t, x, y). t=ν, xu,v=σ iu,iv, y u,v=δ iu,iv In the case where = 0, W p (ν, Σ, 0) is caed the rea centra Wishart distribution and is denoted by W p (ν, Σ). It foows from Remark 3.4 that the moments of the centra rea Wishart distribution are written as specia vaues of Ψ. Coroary 3.7 For W = (w i,j ) W p (ν, Σ) E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = Ψ (t, x) t=ν, xu,v=σ u,v = Ψ (ν, Σ). Coroary 3.8 For W = (w i,j ) W p (ν, Σ) E[w i1,i 2 w i3,i 4 w i2n 1,i 2n ] = E[w i w 1,i 1 i w 2 ] = Ψ (t, x),i 2 iṅ,i n. t=ν, xu,v=σ iu,iv Next we consider directed graphs to describe the moments of compex Wishart distributions. For a directed graph (V, E) and variabes x = (x i,j ), we define the weight monomia x E by x E = (v,u) E x v,u. Let E M(V ). The pair (V, E) is a directed graph whose connected components are directed chains and directed cyces without chords. We define en(e) to be the number of cyces (and sef oops) in (V, E). The vertices V \ start(e) are the endpoints of the chains in (V, E), whie the vertices V \ end(e) are the start points of the chains in (V, E). We define Ě by Ě = (v, u) K V \start(e),v \end(e) There exists a chain from u to v in E. } KV.

5 Formuas moments of Wishart distributions 825 Remark 3.9 For E M(V ), Ě can be defined as a subset of K V satisfying the foowing conditions: Ě M(V ), Ě E =, Ě E P(E), The number of connected components in (V, E) equas the number of connected components in (V, Ě E). Remark 3.10 For E M(V ), we can aso define en(e) by en(e) = (the number of connected components in (V, E)) Ě. Remark 3.11 We can identify E P(V ) with the eement σ E of the symmetric group S n such that σ E (i) = j for each (i, j) E. For each E, en(e) is the number of cyces of σ E. Definition 3.12 For a set K K V of directed edges, we define poynomias Φ K and Ψ K by Φ K (t, x, y) = t en(e) x E yě, Ψ K (t, x) = t en(e) x E. E M(V,K) E P(V,K) We aso respectivey define Φ(t, x, y) and Ψ(t, x) to be Φ KV (t, x, y) and Ψ KV (t, x). Remark 3.13 By definition, Ψ K (t, x) = Φ K (t, x, 0) for each K K V. We describe the moments of compex Wishart distributions as specia vaues of the generating functions. Theorem 3.14 Let W = (w i,j ) be a random matrix distributed according to the compex noncentra Wishart distribution CW p (ν, Σ, ). Then E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = Φ(t, x, y). t=ν, xu,v=σ u, v, y u,v=δ u, v Coroary 3.15 For W = (w i,j ) CW p (ν, Σ, ) E[w i1,i 2 w i3,i 4 w i2n 1,i 2n ] = E[w i w 1,i 1 i w 2 ] = Φ(t, x, y),i 2 iṅ,i n. t=ν, xu,v=σ i u, y,i v u,v=δ i u,i v By substituting 0 for in the theorem, we have the foowing formua for the centra compex case. Coroary 3.16 For W = (w i,j ) CW p (ν, Σ) E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = Ψ(t, x). t=ν, xuv=σ u, v Coroary 3.17 For W = (w i,j ) CW p (ν, Σ) E[w i1,i 2 w i3,i 4 w i2n 1,i 2n ] = E[w i w 1,i 1 i w 2 ] = Ψ(t, x),i 2 iṅ,i n. t=ν, xu,v=σ i u,i v

6 826 Yasuhide NUMATA and Satoshi KURIKI Remark 3.18 For a square matrix A = (a ij ), the α-determinant (or α-permanent) is defined by det α (A) = σ S n α n en(σ) a 1,σ(1) a 2,σ(2) a n,σ(n). This poynomia is an α-anaogue of both the determinant and the permanent. Equivaenty, the α- determinant is nothing but the ordinary determinant and permanent for α = 1 and 1, respectivey. (See aso [16, 17].) Through the identification in Remark 3.10, we have α n Ψ(α 1, A) = det α (A). Moreover, by Coroary 3.16, the moments of the compex centra Wishart distribution are expressed by α-determinants. In [12], Matsumoto introduced the α-pfaffian, which is defined by pf α (A) = E P (V ) ( α) n en(e ) sgn(e )A E for a skew-symmetric matrix A, where sgn(e )A E is defined to be sgn(x)a x 1,x 1 a xṅ,x n for x S 2n such that E = x 1, x 1 },..., x ṅ, x n } }. Since A is skew symmetric, sgn(e )A E is independent from choices of x S 2n. The α-pfaffian is an anaogue of the Pfaffian. Equivaenty, in the case when α = 1, α-pfaffian pf 1 (A) is nothing but the ordinary Pfaffian pf(a), i.e., sgn(x)a x 1 x 1 a xṅx n. Let us define the poynomia hf α (A) by hf α (B) = E P (V ) α n en(e ) B E for a symmetric matrix B. The poynomia hf α (B) is an α-anaogue of the Hafnian. Equivaenty, hf α (B) is the ordinary Hafnian hf(b), i.e., b x 1 x 1 b xṅx n, for α = 1. By definition, hf α (B) = α n Ψ (α 1, B). In this sense, the moments of the rea centra Wishart distributions are expressed by α-hafnians. 4 Appication 4.1 Reation between rea and compex cases There exist bijections between directed graphs and undirected graphs which preserve the weight monomias in some specia cases. These bijections induce equations between weighted generating functions of matchings. From the equations, we can obtain some formuas for the moments of compex and rea Wishart distributions with specia parameters Prototypica case As in Remark 2.1, there exists a correspondence between directed graphs and bipartite graphs. Lemma 4.1 The map M(V, K V ) (u, v) u, v} M (V, K V, V ) is a bijection. The map induces the bijection P(V, K V ) (u, v) u, v} P (V, K V, V ).

7 Formuas moments of Wishart distributions 827 These bijections impy Φ KV (t, x, y) = Φ K (t, x, y) and Ψ V KV (t, x) = Ψ K (t, x). If Σ = (σ, V V, uv) V and = (δ uv) satisfy σ u,v = 0 and δ u,v = 0 for u, v} K V K V, then Φ (t, x, y) t=ν, xu,v=σ u,v, yu,v=δ u,v = Φ K (t, x, y) V, V t=ν, x u,v=σ u,v, yu,v=δ u,v If Σ = (σ uv ) and = (δ uv ) satisfy σ u, v = σ u,v and δ u, v = δ u,v for u, v V, then the equation impies Φ(t, x, y) = Φ (t, x, y). t=ν, xu,v=σ u, v, y u,v=δ u, v t=ν, xu,v=σ u,v, y u,v=δ u,v Hence we have the foowing: Propsition 4.2 Let Σ = (σ u,v), = (δ u,v), Σ = (σ u,v ) and = (δ u,v ) satisfy σ u,v = 0, δ u,v = 0 for u, v} K V K V, and σ u, v = σ u, v, δ u, v = δ u, v for u, v V. For W = (w u,v) CW p (ν, Σ, ) and W = (w u,v) W p (ν, Σ, ), E[w 1, 1 w ṅ, n] = E[w 1, 1 w ṅ, n ] Centra case Next we consider the centra Wishart distribution. In this case, we may consider ony perfect matchings. We define P (V ) and P(V ) by } } P (V ) = (E, ω ), P(V ) = E P(V ), (E, ω) ω : E ±1 }. E P (V ), ω : cyces in (V, E E 0) } ±1 }. Lemma 4.3 There exists a bijection between P (V ) and P(V ). We sha give a bijection ψ between P (V ) and P(V ) in Section The bijection preserves the weight monomias, equivaenty, t en(e) x E = t en(e ) (x ) E for eements E P (V ) corresponding to E P(V ), in the case when x = (x u,v ) and x = (x u,v) satisfy x u,v = x u,v for any u, v V and any u, v } K u v }, u v }. Hence Proposition 4.4 foows from the foowing equations: 2 n Ψ(t, x) = 2 n t en(e) x E = t en(e) x E, Ψ (2t, x) = 2 n E P(V ) E P(V ) (2t) en(e ) x E = t en(e ) x E. E P (V ) E P(V ) Propsition 4.4 Let Σ = (σ u,v ) and Σ = (σ u,v) satisfy σ u,v = σ u,v for any u, v V and any u, v } K u v }, u v }. Then 2 n Ψ(t, x) = Ψ (2t, x) t=ν, xu,v=σ u, v. t=ν, xu,v=σ u,v Coroary 4.5 Let Σ = (σ u,v ) and Σ = (σ u,v) satisfy σ u,v = σ u,v for any u, v V and any u, v } K u v }, u v }. For W = (w u,v ) CW p (ν, Σ) and W = (w u,v) W p (ν, Σ ), E[w 1, 1 w ṅ, n] = E[w 1, 1 w ṅ, n ].

8 828 Yasuhide NUMATA and Satoshi KURIKI Noncentra case Next we consider the noncentra Wishart distribution. In this case, we consider a matchings in compete graphs. We define M(V ) and M (V ) by M (V ) = (E, ω ) } E M (V ), ω : cyces in (V, E E 0 ) } ±1 }, M(V ) = (E, ω) Lemma 4.6 There exists a bijection between M(V ) and M (V ). } E M(V ), ω : E ±1 }. We sha give a bijection between P (V ) and P(V ) in Section The bijection preserves the weight monomia in a specia case. Hence Proposition 4.7 foows from the foowing equations: Φ (2t, x, y) = (2t) en(e ) x E yě = t en(e ) x E yě, where 2x = (2x u,v ) Φ(t, 2x, y) = E M (V ) E M(V ) t en(e) (2x) E yě = (E,ω ) M (V ) (E,ω) M(V ) t en(e) x E yě, Propsition 4.7 Let Σ = (σ u,v ), = (δ u,v ), Σ = (σ u,v) and = (δ u,v) satisfy σ u,v = σ u,v δ u,v = δ u,v for any u, v V and any u, v } K u v }, u v }. Then Ψ(t, 2x, y) = Ψ (2t, x, y). t=ν, xu,v=σ u, v, y u,v=δ u, v t=ν, xu,v=σ u,v, y u,v=δ u,v Coroary 4.8 Let Σ = (σ u,v ), = (δ u,v ), Σ = (σ u,v) and = (δ u,v) satisfy σ u,v = σ u,v δ u,v = δ u,v for any u, v V and any u, v } K u v }, u v }. For W = (w u,v ) CW p (ν, 2Σ, ) and W = (w u,v) W p (2ν, Σ, ), E[w 1, 1 w ṅ, n] = E[w 1, 1 w ṅ, n ] Construction of Bijections Here we construct bijections to prove Lemmas 4.3 and 4.6. First we construct a bijection ψ from P(V ) to P (V ). To define the bijection, we define the foowing map. For (E, ω) P(V ), et h E,ω and h E,ω be maps from V to V defined by v if ω((u, v)) = 1 for some (u, v) E, h E,ω (v) = v otherwise, h v if ω((u, v)) = 1 for some (u, v) E, E,ω(v) = v otherwise. Remark 4.9 For (E, ω) P(V ) and v V, h E,ω (v), h E,ω (v)} E 0. First we construct E P (V ) for each (E, ω) P(V ). For each (E, ω) P(V ), we define a surjection ψ E,ω : E K V, and then we define E to be the image ψ E,ω (E). Let (v 1, v 2 ), (v 2, v 3 ),..., (v k 1, v k ),

9 Formuas moments of Wishart distributions 829 (v k, v 1 ) be a cyce in E such that v 1 = min v 1,..., v k }. For each directed edge in the cyce, we define ψ E,ω by ψ E,ω ((v 1, v 2 )) = v 1, h E,ω (v 2 )}, ψ E,ω ((v i, v i+1 )) = h E,ω(v i+1 ), h E,ω (v i+1 )} (for i = 2,..., k 1), ψ E,ω ((v k, v 1 )) = h E,ω(v k ), v 1 }. Then the image of the cyce forms a cyce C in the undirected graph E E 0. For the cyce C, we define ω (C ) to be ω((v k, v 1 )). Remark 4.10 It is easy to construct the inverse map of ψ, which impies that ψ is bijective. Remark 4.11 This correspondence ψ is equivaent to the one in [4], which is described in more agebraic terms. Let S 2m be the 2m-th symmetric group, and et B m = S m Z/2Z the hyperoctehedra group, i.e., the subgroup of the permutations π S 2m such that π(ṅ) π( n) = 1 for a n = 1,..., m. For gb m S 2m /B m, we can define E gbm by E gbm = g(ṅ), g( n)} n = 1,... m } P(V ), and we can identify eements gb m S 2m /B m with perfect matchings in (V, K V ). Through this identification, the correspondence in Section 4 of [4] is equivaent to ours. Next we construct a bijection ϕ from M(V ) to M (V ). For each (E, ω) M(V ), we sha define (E, ω ) M(V ). For each cyce in E, we construct undirected edges and ω in the same manner as ψ. To define the undirected edges corresponding to chains, we define t E,ω and t E,ω by v if ω((u, v)) = 1 for some (v, u) E, t E,ω (v) = v otherwise, t v if ω((u, v)) = 1 for some (v, u) E, E,ω(v) = v otherwise. Let (v 1, v 2 ), (v 2, v 3 ),...,(v k 1, v k ) be a maxima chain in E. If v 1 < v k, then we define ϕ E,ω by ϕ E,ω ((v 1, v 2 )) = v 1, h E,ω (v 2 )}, ϕ E,ω ((v i, v i+1 )) = h E,ω(v i ), h E,ω (v i+1 )} (for i = 2,..., k 1). If v 1 > v k, then we define ϕ E,ω by ϕ E,ω ((v k 1, v k )) = t E,ω (k k 1 ), v k } ϕ E,ω ((v i 1, v i )) = t E,ω (v i 1 ), t E,ω(v i )} (for i = k 1,..., 2). Then the image of the maxima chains forms a maxima chain in the undirected graph E E 0. Remark 4.12 It is easy to construct the inverse map of ϕ, which impies ϕ is bijective. 4.2 Noncentra chi-square distribution In this section, we consider the Wishart distributions for a specia parameter, which is inked with the noncentra chi-square distributions.

10 830 Yasuhide NUMATA and Satoshi KURIKI Propsition 4.13 Let σ u,v = σ, δ u,v = δ. For W = (w u,v ) CW p (ν, Σ, ), n E[w 1, 1 w 2, 2 w ṅ, n] = g mn ν σ m δ n m, where g mn is the number of E K V such that en(e) =, E = m and Ě = n m. Coroary 4.14 For the noncentra compex chi-square distribution χ 2 ν(δ) with ν degrees of freedom and the noncentraity parameter δ, its n-th moment E(w n ) is given as foows: n E[w n ] = g mn ν δ n m. In the case where we add a new directed edge whose staring point is a fixed vertex, we have just one choice of end-points that increase the number of cyces. Hence we obtain Lemma Lemma 4.15 Let 0 m n. Then the generating function G mn (t) of g mn with respect to the number of cyces satisfies G mn (t) = ( ) n m g mn t = (t + n i). m 0 We aso obtain the foowing coroary, which is we-known expression for the noncentra chi-square distribution (e.g. [5]) Coroary 4.16 For the n-th moment E(w n ) of the noncentra chi-square distribution χ 2 ν(δ) with ν degrees of freedom and the noncentraity parameter δ, n n n ( ) n m E[w n ] = g mn ν δ n m = G mn (ν)δ n m = δ n m (ν + n i). m Remark 4.17 The numbers s n (m, ) defined by the foowing generating function are caed the noncentra Stiring numbers of the first kind: m s n (m, )t = (t + n i). ( If m = n, then s n (m, ) is the Stiring number of the first kind. Lemma 4.15 impies that g mn = n ) m sn (m, ). Equivaenty, we can expicity describe the moments of the noncentra chi-square distribution χ 2 ν(δ) with the noncentra Stiring numbers. Koutras pointed out that moments of some noncentra distributions are described with the noncentra Stiring numbers of the first kind [7]. Next consider the rea case. Propsition 4.18 Let σ u,v = σ, δ u,v = δ. For W = (w u,v ) W p (ν, Σ, ), n E[w 1, 1 w 2, 2 w ṅ, n] = g mnν σ m δ n m, where g mn is the number of E K V such that en(e ) =, E = m and Ě = n m.

11 Formuas moments of Wishart distributions 831 Coroary 4.19 For the noncentra chi-square distribution χ 2 ν(δ) with ν degrees of freedom and the noncentraity parameter δ, its n-th moment E(w n ) is given as: n E[w n ] = g mnν δ n m, where g mn is the number of E K V such that en(e ) =, E = m and Ě = n m. Proposition 4.7 and Lemma 4.15 impy Lemma Lemma 4.20 Let 0 m n, n 0. Then the generating function G mn(t) of g mn with respect to the number of cyces satisfies G mn(t) = ( ) n m g mnt = (t + 2(n i)). m 0 Coroary 4.21 For the n-th moment E(w n ) of the noncentra chi-square distribution χ 2 ν(δ) with ν degrees of freedom and the noncentraity parameter δ, n n n ( ) n m E[w n ] = g mnν δ n m = G mn(ν)δ n m = δ n m (ν + 2(n i)). m 4.3 Bivariate chi-square distribution We can expicity describe the moments of Wishart distributions by enumerating the matchings satisfying some conditions. For exampe, in Proposition 4.23, we obtain the description of the moments of the bivariate rea chi-square distribution, which was introduced by Kibbe [6]. The formuas impy formuas for the compex distribution by Proposition 4.7. See [8] for detais and other appications. Propsition 4.22 Let Σ = (σ uv ) and = (δ uv ) satisfy 1 (u, v 2b or 2b + 1 u, v), σ u,v = ρ (otherwise), For a random matrix W = (w u,v ) W b+c (ν, Σ, ), E[w 1, 1 w ḃ, b w (b+1), (b+1) w (b+c), (b+c) ] δ u,v = 0. min(b,c) = ρ 2a 2 a b!c! a b a c a (ν + a(a i)) (ν + a(b i)) (ν + a(c i)). (b a)!(c a)!a! a=0 ( ) 1 ρ Propsition 4.23 Let Σ =, and W = (w ρ 1 u,v ) W 2 (ν, Σ). For b, c Z 0, E[w b 1,1w c 2,2] = min(b,c) a=0 ρ 2a 2 a b!c! (b a)!(c a)!a! a b a c a (ν + 2(a i)) (ν + 2(b i)) (ν + 2(c i)). Remark 4.24 In [14], Nadarajah and Kotz derived another expression for E[w b 1,1w c 2,2] with the Jacobi poynomias.

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