Distributions on matrix moment spaces

Transcription

1 Distributions on matrix moment spaces Holger Dette, Matthias Guhlich Ruhr-Universität Bochum Fakultät für Mathematik Bochum, Germany Jan Nagel Technische Universität München Fakultät für Mathematik Garching, Germany April 6, 202 Abstract In this paper we define distributions on the moment spaces corresponding to p p real or complex matrix measures on the real line with an unbounded support. For random vectors on the unbounded matricial moment spaces we prove the convergence in distribution to the Gaussian orthogonal ensemble or the Gaussian unitary ensemble, respectively. Keywords and Phrases: moment spaces, random matrices, matrix measures, random moments, Gaussian ensemble, Laguerre ensemble, Wigner s semi circle law, Marchenko-Pastur law AMS Subject Classification: 60F05;5A52;30E05 Introduction In recent years there has been considerable interest in generalizing many of the results on classical moment theory to the case of matrix measures. Wiener and Masani (957) introduced matrix measures on the unit circle in the study of multivariate stochastic processes and their spectral theory. Whittle (963) followed the same approach and established a connection to matrix polynomials, that is, polynomials with matricial coefficients. Already Karlin and McGregor (959) studied a random walk with a doubly infinite transition matrix with help of matrix polynomials, however without special consideration of the matricial structure. Delsarte et al. (978) orthogonalized polynomials with respect to matrix measures on the unit circle. Duran and van Assche (995), Duran (995, 999) and Duran and Lopez-Rodriguez (996) were the first who investigated matrix orthogonal polynomials with respect to matrix measures on the real line and generalized

2 many results from the scalar case to the matrix case. Typical examples include the three-termrecursion, quadrature formulas and ratio asymptotics. Applications to stochastic processes with two-dimensional state space were discussed by Dette et al. (2006), who expressed transition probabilities and the recurrence of states in terms of matrix measures and matrix orthogonal polynomials. In contrast to moment spaces corresponding to (probability) measures the structure of moment spaces corresponding to matrix measures is much richer and not very well understood. In the scalar case Chang et al. (993) investigated a uniform distribution on the moment space corresponding to measures on the interval [0, ]. Their investigation was motivated by the consideration of a typical point in the moment space and they studied the asymptotic properties of random moment vectors with increasing dimension. Gamboa and Lozada-Chang (2004) considered large deviation principles for random moment sequences on this space, while Lozada-Chang (2005) investigated similar problems for moment spaces corresponding to more general functions defined on a bounded set. More recently, Gamboa and Rouault (2009) discussed random spectral measures related to moment spaces of measures on the interval [0, ] and moment spaces related to measures defined on the unit circle. Dette and Nagel (202a) considered distributions on moment spaces corresponding to scalar measures on the real line with an unbounded support. For matrix measures the corresponding moment of a matrix measure is given by a symmetric (hermitian) matrix and Dette and Studden (2002) obtained a characterization of the compact moment space corresponding to matrix measures on a compact interval. Dette and Nagel (202b) used these results to investigate the asymptotic properties of random vectors with values in the moment space corresponding to matrix measures on the interval [0, ]. The aim of the present paper is to get a better understanding of the properties in the non compact case. For this purpose we define probability distributions on matrix moment spaces corresponding to measures with an unbounded support and study their asymptotic behavior with a increasing dimension. The remaining part of this paper is organized as follows. In Section 2 we introduce the basic notation, define distributions on the moment spaces corresponding to matrix measures on unbounded intervals and state our main results. In Section 3 we consider matrix orthogonal polynomials and their relation to moments of matrix measures. In Section 4 we use this relation to prove our main results. Finally in Section 5 we extend these results to matrix moment spaces corresponding to matrix measures with complex entries. Finally some technical details have been deferred to the appendix. 2 Matrix moment spaces Throughout this paper let (S p (R), B(S p (R))) denote the measurable space of all p p symmetric matrices with real entries, where B(S p (R)) is the Borel field corresponding to the Frobenius norm A = tr(a 2 ) on S p (R). The set S + p (R) S p (R) denotes the subset of positive definite matrices 2

3 and for a matrix A S p (R), A is the determinant of A. Let T be a subset of the real line with corresponding Borel field B(T ). A (S p (R)-valued) matrix measure Σ on a measureable space (T, B(T )) is a p p matrix of signed measures on (T, B(T )) such that for all Borel sets A T the matrix Σ(A) is symmetric and nonnegative definite. Additionally we require the matrix measure to be normalized, that is Σ(T ) = I p, where I p denotes the p p identity matrix. We consider on S p (R) the integration operator (2.) dx := i j dx ij, the product Lebesgue measure with respect to the independent entries of a symmetric matrix. For an integrable function f : S p (R) R the integral (2.2) f(x)dx is the iterated integral with respect to each of the elements x ij, i j (see Muirhead (982) or Gupta and Nagar (2000)). The k-th moment of a matrix measure is then defined as (2.3) M k (Σ) := x k dσ(x) for k 0. The set of all R-valued matrix measures on (T, B(T )) for which all moments exist is denoted by P p (T ) and we define the n-th moment space of matrix measures by (2.4) M p,n (T ) := { (M (Σ),..., M n (Σ)) T Σ P p (T ) }. Analogous to the compact case in Dette and Nagel (202b) we obtain a characterization of the moment spaces M p,n ([0, )) and M p,n (R) in terms of Hankel matrices, which are defined for matrices M k S p (R), k 0 as (2.5) H 2m = M 0 M m M M 2 M m M m+.., H 2m =.. M m... M 2m M m M m+... M 2m M 2m, and (2.6) H 2m+ = M M m+. M m+... M 2m+., H 2m+ = M 0 M M m M m+. M m M m+... M 2m M 2m+.. The following lemmas give a characterization of M p,n ([0, )) and M p,n (R). The proof follows by similar arguments as in Dette and Studden (2002) and is therefore omitted. Note that the authors consider non-normalized measures, but the arguments can be extended to matrix probability measures. 3

4 Lemma 2.. A vector of matrices (M,..., M n ) T S p (R) n is an element of the moment space M p,n ([0, )) if and only if for all k n the Hankel matrices H k are nonnegative definite. The vector (M,..., M n ) T is an interior point of M p,n ([0, )) if and only if for all k n the matrices H k are positive definite. We define the vectors of matrix moments and the symmetric matrices h T 2m = (M m+,, M 2m ), h T 2m = (M m,, M 2m ), h T 2m = (M m M m+,, M 2m M 2m ), h T 2m = (M m M m+,, M 2m 2 M 2m ), (2.7) (2.8) Mn+ := h T nh n h n, n, M n+ + := M n h T n H n h n, n 2, where for the sake of completeness we set M = 0 p, M + = M 0 and M + 2 = M. If (M,..., M n ) T is in the interior of M p,n ([0, )), there exist in contrast to the compact case no upper bound for the (n + )-th moment. A vector of symmetric matrices (M,..., M n+ ) T is a moment vector in M p,n+ ([0, )) if and only if (2.9) M n+ M n+ where the matrices are compared with respect to the Loewner ordering. The inequality (2.9) follows because M n+ Mn+ is the Schur complement of M n+ in H n+ and the matrix H n+ is nonnegative definite if and only if the matrix H n and the Schur complement M n+ Mn+ are nonnegative definite. (M,..., M n+ ) T is an interior point of M p,n+ ([0, )) if and only if Mn+ < M n+ holds. For the remaining case of the whole real line, we obtain the following characterization of elements of the corresponding moment space. Lemma 2.2. The vector of matrices (M,..., M n ) T S p (R) n is a moment vector in M p,n (R) if and only if for all k with 2k n the Hankel matrices H 2k are nonnegative definite. The vector (M,..., M n ) T in in the interior of M p,n (R) if and only if for all k with 2k n the Hankel matrices H 2k are positive definite. Consider the moment vector (M,..., M 2n ) T Int M p,2n (R). There exist no bounds (with respect to the Loewner ordering) for the next moment, that is for any M 2n+ S p (R) the matrix vector (M,..., M 2n+ ) T is an interior point in M p,2n+ (R). For the next even moment we can define as in (2.7) the matrix M 2n+2. Then (M,..., M 2n+2 ) T is a moment vector if and only if M 2n+2 M 2n+2 and it is in the interior of M p,2n+2 (R) if and only if M 2n+2 < M 2n+2. 4

5 Now we define a density on the moment space M p,n ([0, )) by n g p,n (γ,δ) (M n ) = c (Mk p,k M k ) (M k M k ) γ k (2.0) k= exp ( δ k tr(m k M k ) (M k M k )) {Mk >M k }, where the parameters satisfy γ k > 2 (p ) and δ k > 0 and we will show in Section 4 that the normalization constant is given by δ pγ k+ 2 p(p+)(n k+) k (2.) c p,k = Γ p (γ k + (p + )(n k + )). 2 Here Γ p denotes the multivariate gamma function, which is defined by Γ p (z) := X z 2 (p+) e tr(x) dx S + p (R) [see Muirhead (982)]. The density g p,n (γ,δ) is the matrix analog of the density on the scalar moment space considered by Dette and Nagel (202a), which in turn is the natural extension of beta-type densities in the compact case. Our next result gives the asymptotic distribution of the vector of the first k components of random matrix moments distributed according to the density g p,n (γ,δ). For this purpose recall that a random symmetric p p matrix is governed by the Gaussian orthogonal ensemble (GOE p ), if its density is given by (2.2) f G (X) = (2π) p/2 π p(p )/4 e 2 tr X2. We define the matricial Marchenko-Pastur distribution η p by x(4 x) (2.3) dη p (x) := {0<x<4} dx I p, 2πx and obtain from the diagonal structure of this measure that the k-th moment satisfies M k (η p ) = c k I p, where c k is the k-th moment of the scalar Marchenko-Pastur distribution, that is, the k-th Catalan number (see e.g. Tulino and Verdú (2004)). Theorem 2.3. Assume that the vector of random moments M n M p,n ([0, )) has a distribution with density g p,n (γ,δ) where δ i = n (n) (p + ) for all i. For k denote by M 2 k the projection of M n onto the first k matrices and let M k (η p ) = (c I p,..., c k I p ) T contain the first k moments of the matricial Marchenko-Pastur distribution η p. Then the convergence n 2 (p + )(C I p )(M (n) D k M k (η p )) X k n holds, where C R k k is a lower triangular matrix with entries C, =... = C k,k =, ( ) ( ) 2i 2i C i,j =, j < i, i j i j and X k = (X,..., X k ) T is a vector of independent, GOE p -distributed random matrices. 5

6 The proof of Theorem 2.3 and Theorem 2.4 below is referred to Section 4, which contains also several results of independent interest. It requires some more detailed explanation of the relation between the moment of a matrix measure and the coefficients of matrix orthogonal polynomials, which will be presented in the following section. For the definition of a density on the moment space M p,n (R) we also need some basic facts about matrix polynomials. A matrix polynomial is a polynomial P (x) = A n x n + A n x n A 0, with coefficients A k R p p. If Σ P p (R) is a matrix measure, we define the (left) inner product on the space of matrix polynomials by ( p ) (2.4) P (x), Q(x) := P (x)dσ(x)q(x) T = P (x) ik Q(x) jl dµ kl (x). k,l= i,j p This inner product is matrix valued and R-linear in both arguments, but it is not a scalar product. The inner product (2.4) was considered by Duran (995) and several applications are discussed in Sinap and van Assche (996). It is possible to construct matrix polynomials P k (x) orthogonal with respect to the inner product,, that is, (2.5) P n (x), P m (x) = 0 p for n m. The degree of a matrix polynomial P n (x) = n k=0 A kx k is n, if A n 0 p and P n (x) is called monic, if A n = I p. In order to construct monic orthogonal polynomials up to degree N with respect to a matrix measure Σ, we assume that the Hankel matrix M 0 M N H 2N 2 =. M N.... M 2N 2 of the moments of Σ is positive definite. By Lemma 2.2 this is equivalent to the fact that (M,..., M 2N 2 ) T Int M p,2n 2 (R). If P n (x) = n k=0 A kx k is a monic matrix polynomial of degree n < N, then for any z R p \ {0}, z T P n (x), P n (x) z =z T (A T 0,..., A T n)h 2n (A T 0,..., A T n) T z =(z T A T 0,..., z T A T n)h 2n (z T A T 0,..., z T A T n) T > 0. This shows that P n (x), P n (x) is positive definite. Then the Gram-Schmidt-procedure can be applied to the matrix monomials I p, xi p, x 2 I p,..., which results in P 0 (x) = I p and recursively (2.6) P n (x) =x n I p x n I p, P n (x) P n (x), P n (x) P n (x)... The assumption that the polynomials P n (x) is monic is indeed necessary, otherwise P n (x), P n (x) would be singular for all vectors z which are in the kernel of all matrices A,..., A n. 6

7 x n I p, P 0 (x) P 0 (x), P 0 (x) P 0 (x) for n N. The basic properties of the inner product yield (2.5) for n m. Note that if H 2N is not positive definite, P N (x), P N (x) may be singular and P N+ (x) can not be defined. In the following discussion we suppose that H 2N is positive definite for all N. As in the scalar case, the monic orthogonal matrix polynomials satisfy a three term recursion (2.7) xp n (x) = P n+ (x) + B n+ P n (x) + A n P n (x), n with matricial recursion coefficients A n, B n+ R p p. By an induction argument, the recursion coefficients can be recursively calculated from (2.8) (2.9) P n (x), x n I p =A n A n... A 0, P n (x), x n+ I p =B n+ A n... A 0 + A n B n A n... A A n... A B A 0 where A 0 = I p, I p = I p = M 0 (see Wall (948) for the scalar versions). Equation (2.8) gives the identity A k = P k (x), P k (x) P k (x), P k (x) and from equation (2.9) we get a recursion for B k. The mapping M 2n (B, A,..., B n ) is even invertible, since equations (2.8) and (2.9) allow a recursive calculation of the moments from the recursion coefficients. For our probabilistic analysis we introduce the symmetrized matrices (2.20) A k (M 2n ) := P k (x), P k (x) /2 P k (x), P k (x) P k (x), P k (x) /2 for k n and for k n, (2.2) B k (M 2n ) := P k (x), P k (x) /2 B k P k (x), P k (x) /2. Here we write A k (M 2n ) and B k (M 2n ) to emphasize the dependence of the recursion coefficients from the moments. Now we define a density on the moment space M p,n (R) by (2.22) h (γ,δ) 2n (M 2n ) = n c Bk exp ( δ 2k tr B k (M 2n ) 2) k= n c Ak A k (M 2n ) γ k exp ( δ 2k tr A k (M 2n )) {Ak >0 p} k= with parameters γ k >, δ k > 0. The normalization constants c Bk and c Ak are given by (2.23) (2.24) c Bk = 2 p/2 ( 2δ2k π ) p(p+)/4, δ pγ k+p(p+)(n k) 2k c Ak = Γ p (γ k + (p + )(n k)). 7

8 Again, this density defines the natural analog of the distributions analysed in the scalar case. The following result gives a central limit theorem for the corresponding vector of random moments. The centering constants are the moments of the matrix-semicircle distribution ρ p defined by (2.25) dρ p (x) := 4 x2 { 2<x<2} dx I p. 2π Theorem 2.4. Assume that the vector of random moments M 2n M p,2n (R) has a distribution with density h (γ,δ) 2n, where the parameters of the density satisfy δ 2i = n(p + ) and δ 2i = (n) n(p + ) for all i. For k fixed denote by M 2 k = (M,..., M k ) the vector of the first k matrices of M 2n. Then n(p + )(D I p )(M (n) D k M k (ρ p )) X k, n where M k (ρ p ) are the first moments of the matrix-semicircle distribution, the matrices in X k = (X,..., X k ) T are independent and GOE p -distributed and D is a k k lower triangular matrix with D i,j = 0 if i + j is odd, D, =... = D k,k = and the remaining entries are given by ( ) ( ) i i D i,j = i j i j. 2 2 Remark 2.5. Dette and Nagel (202b) proved a similar result for random matrix moments uniformly distributed on the space M p,n ([0, ]). It is also possible to obtain results of this type for a more general class of densities f (γ,δ) p,n (M n ) = n k= c p,k (M + k M k ) (M k M k ) γ k (M + k M k ) (M + k M k) δ k {M k <M k<m + k } where γ = (γ k ) k, δ = (δ k ) k are fixed sequences of parameters with γ k, δ k > 2 (p ). If γ k = δ k = 0 for all k, we obtain the uniform distribution on M p,n ([0, ]) considered in this reference. 3 Matrix orthogonal polynomials An important tool for the proof of the results in Section 2 are matrix polynomials. If the support of a matrix measure Σ is a subset of [0, ), it follows by similar arguments as in Dette and Studden (2002) that there exists a sequence of matrices Z n R p p, such that for n the recursion coefficients in (2.7) of the monic matrix polynomials are given by (3.) (3.2) A T n =Z 2n Z 2n, B T n =Z 2n 2 + Z 2n, 8

9 where for the sake of completeness we define Z 0 = 0 p (note that these authors define the inner product in a different way). Throughout this paper we call the variables Z k (and similar quantities) canonical variables. Dette and Studden (2002) also showed the representation (3.3) Z n = (M n M n ) (M n M n ) for n where M0 = 0 p. Since we assume all Hankel matrices H 2N to be positive definite, we have M n > Mn. This section will provide a recursive method to calculate the moments from the recursion coefficients or from the canonical variables Z n. A similar result in the scalar case was shown by Skibinsky (968). We first need to make some definitions. Denote by P(x) T = (P 0 (x) T, P (x) T, P 2 (x) T,... ) the vector of monic orthogonal matrix polynomials and by F (x) T = (I p, xi p, x 2 I p,... ) the vector of matricial monomials. If (3.4) B I p A B 2 I p J = A 2 B 3 I p is the block-tridiagonal matrix of the recursion coefficients, the recursion (2.7) can be written as (3.5) JP(x) = xp(x). The infinite Hankel matrix of the moments of Σ is denoted by (3.6) M = (M i+j ) i,j 0, and let L be the lower block-triangular matrix containing the coefficients of the matrix polynomials, that is (3.7) P(x) = LF (x). Furthermore, R is the shift-operator 0 p I p 0 p... R = 0 p 0 p I p 0 p Although we consider matrices of infinite size, a multiplication of two matrices involves only a finite sum and all products are well-defined. The following Lemma is due to Dette and Nagel (202a) 9

10 Lemma 3.. The matrix L defined by (3.7) is non-singular and the inverse K = L is recursively defined by RK = KJ. Let D = diag(d 0, D,... ) be the block-diagonal matrix with diagonal entries D n = P n (x), P n (x) then the moment matrix M of Σ has the representation M = KDK T. If we write out the block entries in RK = KJ, we see that the blocks of the matrix K can be calculated with the recursion K i+,j = K i,j + K i,j B j+ + K i,j+ A j+, 0 j i where K i,j = 0 p if j < 0 and K i,i = I p (note that the numbering of the blocks in K starts at 0). By the last assertion in Lemma 3., the moment M n is then given by the block in the position n, 0 in the matrix KDK T and therefore M n = K n,0 D 0,0 K 0,0 = K n,0 M 0 In order to obtain a recursion for the matrices Z n, we need the following lemma to express the canonical variables as the recursion coefficients of another measure. The proof is given in the appendix. Lemma 3.2. Let Σ be a matrix measure on [0, ) and Σ s the symmetric matrix measure on R defined by (3.8) Σ s ([ x, x]) = Σ([0, x 2 ]) for all x 0. Then the monic matrix polynomials orthogonal with respect to Σ s satisfy the recursion xp n (x) = P n+ (x) + Z n (Σ) T P n (x), n 0 where Z n (Σ) is the matrix in the recursion of the polynomials orthogonal with respect to Σ. Lemma 3. together with Lemma 3.2 enables us to prove a recursive relation to compute the moments of Σ from the canonical variables Z n. Theorem 3.3. Let Σ P p ([0, )) be a matrix probability measure with canonical variables Z n defined in (3.3). Define the triangular array of matrices G i,j, i, j 0 by G i,j = 0 if i > j, G 0,j = I p and for j i. Then G n,n = M n (Σ). G i,j = G i,j + Z j i+ G i,j 0

11 Proof: For the matrix measure Σ on [0, ) we define the symmetric measure Σ s by (3.8). According to Lemma 3.2 the monic orthogonal polynomials P n (x) of Σ s of even (odd) degree are even (odd) functions. This implies for the block matrix K = (K i,j ) i,j 0 in Lemma 3., which satisfies F (x) = KP(x), that block K i,j is the matrix of zeros if i + j is odd. Furthermore, the blocks in the matrix J of the recursion coefficients are B n = 0 p and A n = Z T n. Here Z n is calculated from the moments of Σ, not of Σ s. The equation RK = KJ gives the recursion (3.9) K i+2j,i = K i+2j,i + K i+2j,i+ Z T i+. Now define G i,j = K T i+j,j i for i j, else G i,j = 0 p, then the matrices G i,j satisfy the asserted recursion. From the equation M = KDK T in Lemma 3. we obtain M n (Σ) = M 2n (Σ s ) = K 0,0 D 0 K T 2n,0 = K T 2n,0 = G n,n. Here we used that Σ is a probability measure and D 0 = M 0 = I p. 4 Weak convergence of random matrix moments 4. Matrix measures on the half-line and proof of Theorem Convergence of canonical variables Dette and Nagel (202b) use symmetrized canonical moments to study random matrix moment on the interval [0, ]. The auxiliary variables in the study of the random moments on the halfline are symmetrized versions of the canonical variables Z k in the recursion of matrix polynomials orthogonal on [0, ). For moments M n = (M,..., M n ) T in the interior of M p,n ([0, )) we define (4.) Z k := (M k M k ) /2 (M k M k )(M k M k ) /2 S + p (R) for k =,..., n. The symmetric matrix Z k is positive definite and similar to Z k. We obtain a mapping (4.2) ψ p,n : { Int M β p,n ([0, )) S + p (R) n M n Z n = (Z,..., Z n ) T, that maps a moment vector to the vector of corresponding canonical variables. If Z k and M,..., M k are given, M k can be calculated with equation (4.). Recursively, all moments M,..., M n can be recovered from canonical variables Z,..., Z n and the mapping ψ p,n is oneto-one. The next Theorem gives the distribution of the canonical variables Z n of the random

12 vector of moments with density g p,n (γ,δ) defined in (2.0). Recall that the distribution of a random variable X S p (R) is given by the Laguerre orthogonal ensemble LOE p (γ, W ) with parameter γ > (p ) and scale matrix W 2 S+ p (R), if it has the density (4.3) f L (X) = Γ p (γ) W γ X γ 2 (p+) e tr W X {X S + p (R)} [see e.g. Muirhead (982)]. If W = I p, we call this distribution central Laguerre orthogonal ensemble and write X LOE p (γ). Theorem 4.. If the vector of random moments M n has a distribution with density g p,n (γ,δ) Z n = (Z,..., Z n ) = ψ p,n (M n ), then the random matrices Z,..., Z n are independent and and Z k LOE p (γ k + (p + )(n k + ), δ k 2 I p) for k =,..., n. Proof: By definition of ψ p,n the canonical variable Z k depends only on M,..., M k, and consequently the Jacobian determinant is obtained as vec(z n ) n vec(m n ) = vec(z k ) vec(m k ) = n M k M k 2 (p+) k= k= n n = Z... Z k 2 (p+) = Z k 2 (p+)(n k). k=2 Here we use the fact, that for X, A S p (R) the determinant of the Jacobi matrix J(Φ A ) of the transformation Φ A : X AXA is given by J(Φ A ) = A p+ [see e.g. Mathai (997)]. The joint density of the canonical variables Z,..., Z n is given by g (γ,δ) Z (Z n ) = k= n c p,k Z k γ k+ 2 (p+)(n k) exp( δ k tr Z k ) {Zk >0 p}, k= and we obtain the distribution according to Definition 4.3. Note that this argument also gives the value of the normalization constant in (2.). The following result provides the weak convergence of the Laguerre orthogonal ensemble if the parameter tends to infinity. Theorem 4.2. Suppose X n LOE p (γ n ) is a sequence of random matrices with parameters γ n. Then it holds that D (X n γ n I p ) GOE p. γn n 2

13 Proof: The moment generating function of X n (i.e., the joint moment generating function of all real random variables in X n ) can be written as a function (4.4) E [ e tr(kxn)] of a symmetric matrix K S p (R) (compare Gupta and Nagar (2000)). By Definition 4.3, E [ e tr(kxn)] = S p + (R) Γ p (γ n ) X γn 2 (p+) e tr X e tr(kx) dx I = I p K γn p K γn X γn 2 (p+) e tr(ip K)X dx = I p K γn Γ p (γ n ) S + p (R) for K < I p. The moment generating function of the standardized random variable is therefore given by E [e ] tr(k γn (X n γ ni p)) = E [e ] tr( γn KX n) e γ n tr K = I p K e γn γ n tr K. γn Let κ,..., κ p denote the eigenvalues of the matrix K, then the moment generating function can be written as p ( κ ) γn p { ( i e γ nκ i = exp γ n log κ ) i } γ n κ i. γn γn i= Now expanding the logarithm yields for I p < K < I p E [e ] { p tr(k γn (X n γ ni p)) = exp γ n k i= k= { } p p = exp 2 κ2 i + k κk 2 k i γn n i= k=3 i= i= ( κi γn ) k γ n κ i } { } ( ) exp 2 κ2 i = exp tr K2. 2 Because exp( tr 2 K2 ) is the moment generating function of the Gaussian orthogonal ensemble the assertion of Theorem 2.3 follows. The random canonical variables Z k are distributed according to the Laguerre orthogonal ensemble and the weak convergence for n follows directly from Theorem 4.2. Note that Z k LOE p (γ k + (p + )(n k + ), δ 2 k I p) implies that δ k Z k LOE p (γ k + (p + )(n k + )) 2 is distributed according to the central Laguerre orthogonal ensemble. The appropriately standardized random matrix δk (Z k I p ) = (δ k Z k δ k ) δk tends in distribution to the Gaussian orthogonal ensemble if δ k depends on n and tends to infinity with the same rate as γ k + (p + )(n k + ). This yields the following result. 2 3

14 Theorem 4.3. Assume that the vector of random moments M n has a distribution with density g p,n (γ,δ) defined in (2.0) and parameters δ k = (p + )n for all k. Then for each component Z 2 k of Z n = (Z,..., Z n ) = ψ p,n (M n ) the weak convergence 2 (p + )n(z D k I p ) GOE p n holds. The non-standardized canonical variable Z k converges under the assumptions of Theorem 4.3 in probability to the identity matrix. Recalling the definition of the matricial Marchenko-Pastur distribution (2.3) and definition (4.) the diagonal structure is carried over to the canonical variables. Therefore we obtain from the scalar case [see Lemma 3.3 in Dette and Nagel (202a)] Z 0 n = ψ p,n (M n (η p )) = ψ,n (c n ) I p = (,..., ) T I p = (I p,... I p ) T. In other words, the random canonical variables Z k converge in the situation of Theorem 4.3 in probability to the corresponding variables of the matricial Marchenko-Pastur distribution η p. By the continuity of the mapping ψp,n, the random moments M k converge to the moments c k I p of the matricial Marchenko-Pastur distribution Proof of Theorem 2.3 For a proof of Theorem 2.3 we need the following concept of differentiability on the level of matrices. Definition 4.4. Let O be an open subset of S p (R) n. A mapping F : O (R p p ) m is called matrix differentiable at X 0, if there exists a matrix L R mp np, such that F (X 0 + H ) F (X 0 ) = LH + o( H ) as H 0. In this case we define the matrix derivative of F at X 0 as F (X 0 ) = F X (X 0 ) = L. Matrix differentiability is a more restrictive form of Fréchet differentiability (for more details on Fréchet differentiability, we refer to Averbukh and Smolyanov (968)). In fact, we require the linear mapping which is the derivative in the Fréchet sense to be a left multiplication, so that we can apply a product rule. If F, G : O R p p are matrix differentiable at X 0, G(X 0 ) = xi p, then F G is matrix differentiable at X 0 and (F G) (X 0 ) = G(X 0 )F (X 0 ) + F (X 0 )G (X 0 ). Now the assertion of Theorem 2.3 is proven analogously to Theorem 2.2 in Dette and Nagel (202b) and follows from Theorem 4.3 and Lemma 4.5 below, which shows that the inverse of the mapping ψ p,k is matrix differentiable. The matrix derivative is the Kronecker product of the derivative in the scalar case with the identity matrix. 4

15 Lemma 4.5. The mapping ψ p,k : S+ p (R) k Int M p,k ([0, )) with ψ p,k (Z k) = M k is matrix differentiable at Z 0 k = (I p,..., I p ) T with derivative Proof: For r k consider the mapping (ψ p,k ) (Z 0 k) = C I p. F r : S + p (R) k R p p, F r (Z) = Z r, which maps the symmetric canonical variables onto the r-th non-symmetric canonical variable. We have the relation Z r = (M r M r ) /2 Z r (M r M r ) /2. Now arguments similar to those in the proof of Theorem 4.5 in Dette and Nagel (202b) show Z r Z (Z0 ) = e T r I p = z r z (z 0 ) I p. This means that the matrix derivative is the Kronecker product of the derivative in the scalar case with the identity matrix. By Lemma 3.2, the moments are sums of products of canonical variables and so the product rule implies the same structure for the derivative of the mapping ψ p,k, which yields ψ p,k Z (Z0 k) = M k Z (Z0 k) = m k z (z 0 k ) I p, and the assertion follows from the calculations in the scalar case (see Dette and Nagel (202a)). 4.2 Matrix measures on R and proof of Theorem Convergence of recursion coefficients In this section we prove the result for the moment space corresponding to matrix measures Σ P p (R). In this case the auxiliary variables A k and B k are the symmetrized versions of the recursion coefficients of orthogonal matrix polynomials defined in equations (2.20) and (2.2), respectively. Note that A k S p (R) and A k > 0 p which follows from the symmetry of the inner product in (2.20) and from the fact that P k (x), P k (x) is positive definite. The following Lemma is proven in the appendix and shows the symmetry of the recursion variables B k as well. Note that the matrices A k and A k and the matrices B k and B k are similar. Lemma 4.6. The matrix B k defined in (2.2) is symmetric, that is B k S p (R). 5

16 With definitions (2.20) and (2.2) we construct the continuous mapping (4.5) ξ p,2n : { Int Mp,2n (R) (S p (R) S + p (R)) n S p (R) M 2n R 2n = (B, A, B 2,..., B n ) T. The symmetric recursion coefficients are in a one-to-one correspondence with the non-symmetric coefficients and the mapping ξ p,2n is one-to-one (Note that A = A and P n (x), P n (x) = A... A n.) Theorem 4.7. Assume that the vector of random moments M 2n Int M p,2n (R) has a distribution with density h (γ,δ) 2n. Then the random recursion coefficients B, A,..., B n are independent and 2δ2k B k GOE p, δ 2k A k LOE p (γ k + (p + )(2n 2k)). 2 Proof: By the equations (2.9) and (2.2) the recursion coefficient B k depends only on the moments M,..., M 2k. The random variable A k depends only on M,... M 2k by identity (2.20). The Jacobi matrix of ξ p,2n is therefore a lower block-triangular matrix with determinant ( vec(ξ p,2n (M 2n )) n ) ( vec(m 2n ) = vec(b k ) n ) vec(a k ) vec(m 2k ) vec(m 2k ). We obtain from (2.20) k= A k = P k (x), P k (x) /2 M 2k P k (x), P k (x) /2 +R, where R is independent from M 2k and therefore (see Mathai (997)) vec(a k ) vec(m 2k ) = P k (x), P k (x) 2 (p+) = A k... A 2 (p+) = A k... A 2 (p+). Rearranging equation (2.9) and an application of (2.2) yields for the other recursion coefficient B k = P k (x), P k (x) /2 B k P k (x), P k (x) /2 = P k (x), P k (x) /2 M 2k (A k... A ) P k (x), P k (x) /2 +R 2 = P k (x), P k (x) /2 M 2k P k (x), P k (x) /2 +R 2, with a matrix R 2 not depending on M 2k, which gives vec(b k ) vec(m 2k ) = P k (x), P k (x) 2 (p+) = A k... A 2 (p+). k= 6

17 This gives for the Jacobian determinant of ξ p,2n ( vec(ξ p,2n (M 2n )) n ) ( vec(m 2n ) = n ) A k... A 2 (p+) A k... A 2 (p+) k=2 k=2 ( n ) ( n ) = A k 2 (p+)(n k) A k 2 (p+)(n k) k= n = k= The recursion coefficients have the joint density h (γ,δ) R (R 2n ) = A k 2 (p+)(2n 2k ). n c Bk exp ( ) δ 2k tr Bk 2 k= n k= c Ak A k γ k+ 2 (p+)(2n 2k ) exp ( δ 2k tr A k ) {Ak >0 p}. k= The product structure of the density yields the independence of the recursion parameters and the density of 2δ 2k B k is given by c Bk 2δ 2k I p 2 (p+) exp ( 2 tr B2 k) = cbk 2δ2k 2 p(p+) exp ( 2 tr B2 k). Consequently 2δ 2k B k GOE p and c Bk = 2δ 2k 2 p(p+) 2π 2 (p+) p = 2 p/2 ( 2δ2k π ) p(p+)/4. The distribution of A k is the non-central Laguerre orthogonal Ensemble LOE p (γ k + (p + )(n k), δ 2k I p), which implies δ 2k A k LOE p (γ k + (p + )(n k)) and c Ak = ( Γ p (γ k + (p + )(n k)) δ 2k I p γ k+(p+)(n k) ) = (Γ p (γ k + (p + )(n k))) δ pγ k+p(p+)(n k) 2k. This proves Theorem 4.7 and the identities (2.23) and (2.24). Theorem 4.8. If in the situation of Theorem 4.7 the parameters satisfy δ 2k = (p + )n and δ 2k = 2 (p + )n, then we have for the kth recursion coefficient B k (p + )nbk GOE p and the kth rescaled recursion coefficient A k converges in distribution to the Gaussian orthogonal Ensemble, that is D (p + )n(ak I p ) GOE p. n 7

18 Theorem 4.8 is a direct consequence from the distribution in Theorem 4.7. The weak convergence of A k follows from the convergence of the Laguerre orthogonal ensemble in Theorem 4.2. Under the assumptions of Theorem 4.8 the matrix vector R (n) k, which contains the first k matrix entries of R 2n = (B, A,..., B n ) T, converges for k fixed in probability to R 0 k, which is the projection of R 0 = (0 p, I p, 0 p, I p,... ) T onto the first k matrix entries. It follows directly from the scalar case in Dette and Nagel (202a), that R 0 k = ξ p,k (M 0 k ), where M 0 k is the projection of M 0 = (0 p, c I p, 0 p, c 2 I p,... ) T onto the first k matrices (the generalization of definition (4.5) to an even number of matricial arguments is straightforward). The vector M 0 contains the moments of the semicircle distribution ρ multiplied with the identity matrix, therefore M 0 = M (ρ p ) is the moment vector of the matrixsemicircle distribution Proof of Theorem 2.4 The proof of Theorem 2.4 now follows as the proof of Theorem 2.3 from the following Lemma and the weak convergence of the random recursion coefficients in Theorem 4.8. Lemma 4.9. The mapping ξ p,k with ξ p,k (R k) = M k is matrix differentiable at R 0 k with derivative (ξ p,k ) (R 0 k) = D I p. Proof: A similar argumentation as in the proof of Lemma 4.5 shows for 2i k and A i R (R0 k) = e T 2i I p = a i r (r 0 k ) I p B i R (R0 k) = e T 2i I p = b i r (r 0 k ) I p for 2i k. From the structure of these derivatives and Lemma 3. we conclude that the derivative of M i with respect to the recursion coefficients is the Kronecker product of the derivative in the scalar case with the identity matrix and ξ p,k R (R0 k) = M k R (R0 k) = m k r (r 0 k ) I p = D I p. 8

19 5 Complex random moments To a large extend, the case of complex matrix measures can be treated analogously to the case of real matrix measures. For the sake of brevity we state only the results and omit the proofs. The integration operator changes on the space of hermitian p p matrices S p (C) to dx = p dx ii drex ij dimx ij, i= i<j that is, we integrate with respect to the p 2 independent real entries of a hermitian matrix. The kth moment of a complex matrix measure on T is defined as M k (Σ) := x k dσ(x) S p (C), T where S p (C) denotes the space of p p hermitian matrices. The complex nth moment space is denoted by M 2 p,n(t ). 5. Random moments of complex matrix measures on the half-line We define the density on the complex moment space M 2 p,n([0, )) as in the real case by (5.) g (γ,δ) p,n (M n ) = n c p,k (M k M k ) (M k M k ) γ k k= exp ( δ k tr(m k M k ) (M k M k )) {Mk >M k }, where the parameters satisfy γ k > p and δ k > 0 and the normalization constant is given by (5.2) c p,k = Γ (2) δ pγ k+p 2 (n k+) k p (γ k + 2(n k + )). The canonical variables Z k in (3.3) are well defined and as in the real case the symmetrized version is given by (5.3) Z k := (M k M k ) /2 (M k M k )(M k M k ) /2 S + p (C) for k =,..., n. The Hermitian matrix Z k is positive definite and similar to Z k. We obtain a mapping (5.4) ψ p,n : { Int M 2 p,n ([0, )) S + p (C) n M n Z n = (Z,..., Z n ), 9

20 that maps a moment vector to the vector of corresponding canonical variables. Recall that the distribution of a random variable X S p (C) is given by the Laguerre unitary ensemble LUE p (γ, W ) with parameter γ > p and scale matrix W S + p (C), if it has the density f L (X) = Γ (2) p (γ) W γ X γ p e tr W X {X S + p (C)}. If W = I p, we call this distribution central Laguerre unitary ensemble and write X LUE p (γ). Γ (2) p is the complex multivariate gamma function and given by Γ (2) p (z) := X z p e tr X dx. S + p (C) Proceeding as in Section 4 gives the following result. Theorem 5.. Assume that the vector of random moments M n has a distribution with density g p,n (γ,δ) and Z n = (Z,..., Z n ) = ψ p,n (M n ), then the random matrices Z,..., Z n are independent and for k =,..., n. Z k LUE p (γ k + p(n k + ), δ k I p) We proceed with asymptotic results for the random matricial moments in M 2 p,n([0, )) for n. The following theorem provides the weak convergence of the Laguerre unitary ensemble if the parameter tends to infinity. For this purpose recall that a random variable X taking values in S p (C) is distributed according to the Gaussian unitary ensemble GUE p, if it has the density f G (X) = (2π p ) p/2 π p(p )/2 e 2 tr X2. Theorem 5.2. Suppose X n LUE p (γ n ) is a sequence of random matrices with parameters γ n. Then it holds that D (X n γ n I p ) GUE p. γn n The remaining arguments in Section 4 stay essentially unchanged, which yields the following result on the weak convergence of random complex moments. Theorem 5.3. Assume that the vector of random moments M n M 2 p,n([0, )) has a distribution with density g p,n (γ,δ) and parameters δ i = np for all i. For k fixed denote by M (n) k the projection of M n onto the first k matrices and assume that M k (η p ) = (c I p,..., c k I p ) contains the first k moments of the matricial Marchenko-Pastur distribution η p. Then the convergence np(c I p )(M (n) D k M k (η p )) X k n holds, where C R k k is the lower triangular matrix in Theorem 2.3 and X k = (X,..., X k ) is a vector of independent, GUE p -distributed random matrices. 20

21 5.2 Random moments of complex matrix measures on R The density on M 2 p,2n (R) in the complex case is given by (5.5) h (γ,δ) 2n (M 2n ) = n c Bk exp ( δ 2k tr B k (M 2n ) 2) k= n c Ak A k (M 2n ) γ k exp ( δ 2k tr A k (M 2n )) {Ak >0 p} k= with parameters γ k >, δ k > 0 and the normalization constants c Bk and c Ak are given by (5.6) (5.7) c Bk = 2 p/2 c Ak = Γ (2) p ( ) p 2 /2 2δ2k π δ pγ k+p 2 (2n 2k) 2k (γ k + p(2n 2k)) Theorem 5.4. Assume that the vector of random moments M 2n Int M 2 p,2n (R) has a distribution with density h (γ,δ) 2n. Then the random recursion coefficients B, A,..., B n are independent and 2δ2k B k GUE p, δ 2k A k LUE p (γ k + p(2n 2k)). Theorem 5.5. If in the situation of Theorem 5.4 the parameters satisfy δ 2k = 2np and δ 2k = np, then we have for the recursion coefficient B k 2npBk GUE p and the rescaled recursion coefficients A k converge in distribution to the Gaussian unitary ensemble, D 2np(Ak I p ) GUE p. n The following result completes this section and gives a central limit theorem for the complex random moments. Theorem 5.6. Assume that the vector of random moments M 2n M 2 p,2n (R) has a distribution with density h (γ,δ) 2n, where the parameters satisfy δ 2i = 2np and δ 2i = np for all i. For k fixed denote by M (n) k the vector of the first k matrices in M 2n. Then 2np(D I p )(M (n) D k M k (ρ p )) X k, n where M k (ρ p ) are the first moments of the matrix-semicircle distribution, X k = (X,..., X k ) is a vector of independent and GUE p -distributed random variables and D is the k k lower triangular matrix in Theorem

22 6 Appendix 6. Proof of Lemma 3.2 The monic polynomials orthogonal with respect to Σ s satisfy a recursion as in (2.7). Equation (2.6) and a simple induction argument yields that P 2n (x) is an even function and P 2n (x) is an odd function and consequently B n = 0 p for all n. For the other recursion coefficients we have A T n = (M 2n 2 (Σ s ) M 2n 2 (Σ s ) ) (M 2n (Σ s ) M 2n (Σ s ) ). A suitable approximation of the identity by step functions shows that the even moments of Σ s are given by M 2n (Σ s ) = x 2n dσ s (x) = x n dσ(x) = M n (Σ). Therefore M n (Σ) M 2n (Σ s ) and M 2n (Σ s ) M n (Σ), which implies M 2n (Σ s ) = M n (Σ). We get for the recursion coefficient A T n = (M n (Σ) M n (Σ) ) (M n (Σ) M n (Σ) ) = Z n (Σ). 6.2 Proof of Lemma 4.6 Denote by C (n) n the coefficient of x n in P n (x), then by the recursion (2.7) we have C (k) k = B k + C (k ) k 2. The construction of the matrix orthogonal polynomials in (2.6) yields C (k) k = xk I p, P k (x) P k (x), P k (x). Another application of the recursion gives for the recursion coefficient B k =C (k ) k 2 C (k) k = x k I p, P k (x) P k (x), P k (x) x k I p, P k 2 (x) P k 2 (x), P k 2 (x) = x k I p, xp k (x) P k (x), P k (x) x k I p, P k 2 (x) P k 2 (x), P k 2 (x) = ( x k I p, P k (x) + x k I p, P k (x) B T k + x k I p, P k 2 (x) A T k P k (x), P k (x) x k I p, P k 2 (x) P k 2 (x), P k 2 (x) = P k (x), P k (x) B T k P k (x), P k (x) 22 )

23 The last equation implies B k = P k (x), P k (x) /2 B k P k (x), P k (x) /2 = P k (x), P k (x) /2 Bk T P k (x), P k (x) /2 =Bk T. Acknowledgements. The work of the authors was supported by the Sonderforschungsbereich Tr/2 (project C2, Fluctuations and universality of invariant random matrix ensembles). References Averbukh, V. and Smolyanov, O. (968). The various definitions of the derivative in linear topological spaces. Russ. Math. Surv., 23:67 3. Chang, F. C., Kemperman, J. H. B., and Studden, W. J. (993). A normal limit theorem for moment sequences. Ann. Probab., 2: Delsarte, P., Genin, Y. V., and Kamp, Y. G. (978). Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circuits Syst., 25: Dette, H. and Nagel, J. (202a). Distributions on unbounded moment spaces and random moment sequences. To appear in: Annals of Probability. Dette, H. and Nagel, J. (202b). Matrix measures, random moments and gaussian ensembles. Journal of Theoretical probability. Dette, H., Reuther, B., Studden, W. J., and Zygmunt, M. (2006). Matrix measures and random walks with a block tridiagonal transition matrix. SIAM J. Matrix Anal. Appl., 29:7 42. Dette, H. and Studden, W. J. (2002). Matrix measures, moment spaces and Favard s theorem on the interval [0; ] and [0; ). Linear Algebra Appl., 345: Duran, A. J. (995). On orthogonal polynomials with respect to a positive definite matrix of measures. Canad. J. Math., 47:88 2. Duran, A. J. (999). Ratio asymptotics for orthogonal matrix polynomials. J. Approximation Theory, 00: Duran, A. J. and Lopez-Rodriguez, P. (996). Orthogonal matrix polynomials: zeros and Blumenthal s theorem. J. Approximation Theory, 84:

24 Duran, A. J. and van Assche, W. (995). Orthogonal matrix polynomials and higher-order recurrence relations. Linear Algebra Appl., 29: Gamboa, F. and Lozada-Chang, L. V. (2004). Large deviations for random power moment problem. Ann. Probab., 32: Gamboa, F. and Rouault, A. (2009). Canonical moments and random spectral measures. J. Theoret. Probab., DOI 0.007/s Gupta, A. K. and Nagar, D. K. (2000). Matrix variate distributions. Capman & Hall/CRC, Boca Raton. Karlin, S. and McGregor, J. (959). Random walks. Illinois J. Math., 3:66 8. Lozada-Chang, L. V. (2005). Asymptotic behavior of moment sequences. Electron. J. Probab., 0: Mathai, A. (997). Jacobians of Matrix Transformations and Functions of Matrix Argument. World Scientific Publ., Singapore. Muirhead, R. J. (982). Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York. Sinap, A. and van Assche, W. (996). Orthogonal matrix polynomials and applications. J. Comput. Appl. Math., 66: Skibinsky, M. (968). Extreme nth moments for distributions on [0, ] and the inverse of a moment space map. J. Appl. Probab., 5: Tulino, A. M. and Verdú, S. (2004). Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory, now Publishers Inc., Hanover, MA. Wall, H. S. (948). Analytic Theory of Continued Fractions. D. van Nostrand Company, Inc. XIII, New York. Whittle, P. (963). On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix. Biometrika, 50: Wiener, N. and Masani, P. (957). The prediction theory of multivariate stochastic processes. Acta Math., 98: