Distributions on unbounded moment spaces and random moment sequences

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1 Distributions on unbounded moment spaces and random moment sequences Jan Nagel Ruhr-Universität Bochum Faultät für Mathemati Bochum, Germany Holger Dette Ruhr-Universität Bochum Faultät für Mathemati Bochum, Germany July 16, 010 Abstract In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marcheno-Pastur distribution and Wigner s semi-circle law. Keyword and Phrases: Gaussian ensemble, Laguerre ensemble, Jacobi ensemble, random matrix, moment spaces, canonical moments, random moment sequences, Wigner s semicircle law, Marceno-Pastur distribution AMS Subject Classification: 60F05, 30E05, 15B5 1 Introduction For a set T R let P(T denote the set of all probability measures on the Borel field of T with existing moments. For a measure µ P(T we denote by m (µ = x µ(dx ; = 0, 1,,... T 1

2 the -th moment and define (1.1 M(T = { m(µ = (m 1 (µ, m (µ,... T µ P(T } R N. as the set of all moment sequences. We denote by Π n (n N the canonical projection onto the first n coordinates and call (1. M n (T = Π n (M(T R n. the n-th moment space. Moment spaces of the form (1.1 and (1. have found considerable interest in the literature [see Karlin and Studden (1966]. Most authors concentrate on the classical moment space corresponding to measures on the interval [0, 1] [see Karlin and Shapeley (1953, Krein and Nudelman (1977, among others]. Chang et al. (1993 equipped the n-th moment space M n ([0, 1] with a uniform distribution in order to understand more fully its shape and the structure. In particular, these authors proved asymptotic normality of an appropriately standardized version of a projection Π (m n of a uniformly distributed vector m n on M n ([0, 1]. Since this seminal paper, several authors have extended these investigations in various directions. Gamboa and Lozada-Chang (004 considered large deviation principles for random moment sequences on the space M n ([0, 1], while Lozada-Chang (005 investigated similar problems for moment spaces corresponding to more general functions defined on a bounded set. More recently, Gamboa and Rouault (009 discussed random spectral measures related to moment spaces of measures on the interval [0, 1] and moment spaces related to measures defined on the unit circle. The present paper is devoted to the problem of defining probability distributions on unbounded moment spaces. We will investigate these distributions from several perspectives. In Section we introduce a class of general distributions on the moment space corresponding to measures defined on a compact interval. By a limiting argument we will derive canonical distributions on the moment spaces corresponding to measures on the unbounded intervals [0, and R, respectively. In Section 3 we show that these distributions appear naturally in the study of random spectral measures of the classical Jacobi, Laguerre and Gaussian ensemble. Finally, in Section 4 we consider random moment sequences distributed according to the new probability distributions on the unbounded moment spaces. In particular, we prove wea convergence of a centered random moment vector, where the centering vector corresponds to the moments of the Marcheno-Pastur law (in the case of the moment space M([0, and to the semi-circle law (for the moment space M(R. Distributions on unbounded moment spaces.1 Canonical moments and recurrence coefficients of orthogonal polynomials Chang et al. (1993 considered random vectors on the moment space M n ([0, 1] governed by a

3 uniform distribution. In the present section we will define a class of more general distributions on the n-th moment space M n ([a, b] corresponding to the set P([a, b] of all probability measures on the interval [a, b]. The motivation for considering this class is twofold. One the one hand we want introduce distributions on the moment space M ([a, b], which are different from the uniform distribution. On the other hand we want to define distributions on unbounded moment spaces as limits of distributions on M ([a, b], when b a. For these purposes we will mae extensive use of the theory of canonical moments, which is briefly defined here for the sae of a self contained presentation. For details we refer to the monograph of Dette and Studden (1997. Let m 1 =(m 1,..., m 1 T M 1 ([a, b] be a given vector of moments of a probability measure on the interval [a, b], then these first 1 moments impose bounds on the -th moment m such that the moment vector m = (m 1,..., m 1, m T is an element of the -th moment space M ([a, b]. More precisely, define for m 1 M 1 ([a, b] m {m = min b } (µ µ P([a, b] with t i dµ(t = m i for i = 1,..., 1, a m + {m = max b } (µ µ P([a, b] with t i dµ(t = m i for i = 1,..., 1, a then it follows that m = (m 1,..., m T Int M ([a, b] if and only if m < m < m +, where Int C denotes the interior of a set C R. Consequently, we define for a point m Int M ([a, b] the canonical moment of order l = 1,..., as (.1 p l = p l (m = m l m l m + l m l ; l = 1,...,. Note that for m Int M ([a, b] we have p l (0, 1; l = 1,..., ; and that p describes the relative position of the moment m in the set of all possible -th moments with fixed moments m 1,..., m 1. It can also be shown that the canonical moments do not depend on the interval [a, b], that is they are invariant under linear transformations of the measure [see Dette and Studden (1997]. Moreover, the definition (.1 defines a one-to one mapping (. ϕ n : { Int M [a,b] n (0, 1 n m n p n = (p 1,..., p n T from the interior of the moment space M [a,b] n onto the open cube (0, 1 n. It can be shown that for a point (m 1,..., m n 1 Int M n 1 ([a, b] the canonical moments appear in the three-term recurrence relation (.3 xp (x = P +1 (x + b +1 P (x + a P 1 (x = 1,..., n 1, (P 0 (x = 1, P 1 (x = x b 1 of the monic orthogonal polynomials m 0 m 1 1 / m 0 m 1 P (x = (.4. ; = 1,..., n m m m 1 x m 1 m 3

4 associated with the vector (m 1,..., m n 1 [see Chihara (1978]. These polynomials are orthogonal with respect to every measure with first moments m 1,..., m n 1 and the recursion coefficients in (.3 are given by (.5 (.6 b +1 = a + (b a((1 p 1 p + (1 p p +1 ; = 0,..., n 1 a = (b a (1 p p 1 (1 p 1 p ; = 1,..., n 1 where we put p 1 = p 0 = 0 (note that a > 0; = 1,..., n. In the case T = [0, the upper bound m + does not exist, but we can still define for a point m 1 Int M 1 ([0, the lower bound m {m = min } (µ µ P([0, with t i dµ(t = m i for i = 1,..., 1, 0 where m = (m 1,..., m T Int M ([0, if and only if m > m of the canonical moments are defined by the quantities. In this case, the analogues (.7 z l = m l m l m l 1 m l 1 l = 1,..., (with m 0 = 0 and related to the coefficients in the three-term recurrence relation (.3 for the monic orthogonal polynomials by (.8 (.9 a = z 1 z, b = z + z 1. Note that (.7 defines a one to one mapping (.10 { Int Mn ([0, (0, n ψ n : m n z n = (z 1,..., z n T from the interior of the moment space M n ([0, onto (R + n. Finally in the case T = R neither m nor m+ can be defined. Nevertheless, there exists also a one to one mapping (.11 ξ n : { Int Mn 1 (R (R R + n 1 R m n 1 (b 1, a 1,..., a n 1, b n T. from the interior of the (n 1th moment space onto the space of coefficients in the three term recurrence relation (.3, which can be considered as the analogue of (.10 and is defined by (.1 x P (xdµ(x = a 1... a = 1,..., n 1, R (.13 x +1 P (xdµ(x = a 1... a (b b +1 = 0,..., n 1, R 4

5 [see for example Wall (1948]. It should be mentioned that the relations for the coefficients in the three term recurrence relation (.3 could be read bacwards, because any measure with support on an interval [a, b] or on the interval [0, is also a measure on the real line. In other words, a measure on the real line is supported on the interval [0, if and only if the coefficients in the recurrence relation (.3 for the corresponding monic orthogonal polynomials allow a representation of the form (.8 and (.9 with non-negative values z. Similarly, the measure is supported on a compact interval if and only if a representation of the form (.5 and (.6 holds, where the quantities p vary in the interval [0, 1]. In the following sections we will use the canonical moments and corresponding quantities on the interval [0, and the real line for the definition of distributions on the corresponding moment spaces.. Distributions on unbounded moment spaces In Section 4 we will show that some of the results of Chang et al. (1993 and Gamboa and Lozada- Chang (004 hold for a rather broad class of distributions on the moment space M n ([a, b]. For the definition of this class, let for 1 f : (0, 1 R be a non-negative integrable function with 1 0 f (xdx > 0, then a distribution on the interior of the moment space M n ([a, b] is defined by (.14 f n (m n = c,n f (p (m n 1 {m <m <m + }, where p (m n is the -th canonical moment defined in (.1 and c,n = 1 1 ((b a n(n+1/ f (x(x x dx n ( = 1,..., n 0 are normalizing constants such that f n is a density on M n ([a, b] (see the proof of the following Theorem.1. Our first Theorem gives the distribution of the canonical moments corresponding to the random vector m n with density f n defined in (.14. Theorem.1. Suppose that m n is a random vector on the moment space M n ([a, b] with density f n defined in (.14. Then the canonical moments p 1 (m n,..., p n (m n are independent and p (m n has the density for 1 n. c,n (b a n(n+1/ f (x(x x n 1 {0<x<1} 5

6 Proof: The -th canonical moment p depends only on the first moments m 1,..., m, which implies for the Jacobian determinant of the mapping ϕ n defined in (. (.15 ϕ n n = m n p (m n m = (m + m 1 = 1 (b a (p i (1 p i 1, i=1 where we have used Theorem and equation (1.3.6 in Dette and Studden (1997 for the last equality. Therefore, the Jacobian determinant in (.15 simplifies as ϕ n n 1 1 = ((b a p i (1 p i = (b a n(n+1/ m n i=1 ( p (1 p (n considering the product structure of f n, this gives the asserted distribution. In Section 4 we will show that the first components of a random vector with density (.14 converge wealy (after appropriate standardization to a normal distribution. This generalizes the results of Chang et al. (1993, who considered the case f 1 for all 1. For the construction of distributions on the unbounded moment space M n ([0,, a special case will be of particular interest, that is f (x = x γ (1 x δ, where γ = (γ N, δ = (δ N are sequences of real parameters, such that γ, δ > 1 for all 1. In this case the density on the moment space M n ([a, b] is given by (.16 f n (γ,δ (m n = c [a,b] n ( m m m + m γ ( m + m δ m + 1 {m m <m <m + }, where (.17 c [a,b] n = {(b a n(n+1/ 1 0 } 1 x n +γ (1 x n +δ dx is the normalizing constant. The choice of the density (.16 is motivated by results of Dette and Studden (1995 who showed that the empirical distribution of the (appropriately normalized roots of the Jacobi polynomials P (γ,δ (x converges wealy to a distribution with unbounded support if γ or δ. Note that for the density f n (γ,δ the canonical moment p has a Beta distribution Beta(γ + n + 1, δ + n + 1. In the following we use densities of the form (.16 to construct a distribution on the unbounded moment space (.18 M n ([0, = { m n (µ = (m 1 (µ,..., m n (µ T µ P([0, }. For this purpose, recall that the relation (.7 defines a one to one mapping between the moment space Int M n ([0, and (R + n. 6

7 Theorem.. Let f (γ(d,δ (d n denote the density defined in (.16 on the moment space M n ([0, d] corresponding to the probability measures on the interval [0, d], where the parameter sequences γ (d = (γ (d N, δ (d = (δ (d γ (d N depend on length d and satisfy d γ > 1, δ (d d d δ R + for all 1. Then for d the density f (γ(d,δ (d n converges point-wise to the function ( g n (γ,δ (m n = c [0, m m γ ( m m (.19 n m 1 m exp δ 1 m 1 m 1 {m >m } 1 = c n [0, z (m n γ exp( δ z (m n 1 {z (m n>0}, where the constant c [0, n Moreover, g (γ,δ n is given by c [0, n = δ γ +n +1 Γ(γ + n + 1. defines a density on the unbounded moment space M n ([0,. Proof: The fact that g n (γ,δ is a density is evident from the transformation in the proof of Theorem.3 below, we prove here only the convergence. For a fixed point m n M n ([0,, there exists a d 0 N with m n M n ([0, d] for all d d 0. Let p n (m n denote the vector of canonical moments corresponding to the vector m n in the moment space M n ([0, d]. We will show at the end of this proof that p (m n = z (m n (.0 (1 + o(1, = 1,..., n, d where the quantities z (m n are defined in (.7. Observing this representation and the definition (.16, it follows for d ( (d γ ( f (γ(d,δ (d n (m n = c [0,d] z (m n n 1 z (d δ (m n (1 + o(1 d d = d (γ(d 1 + +γ(d n c [0,d] z (m n γ exp( δ z (m n (1 + o(1. n Finally, we obtain from (.17 for the normalizing constant d (γ(d 1 + +γ(d n c [0,d] n = d n(n+1/ (γ(d 1 + +γ(d n = d n(n+1/ (γ(d 1 + +γ(d n = c [0, n (1 + o(1, 7 Γ(γ (d Γ(γ (d + δ (d + n + + n + 1Γ(δ (d + n + 1 (δ (d γ +n +1 (1 + o(1 Γ(γ + n + 1

8 which proves the assertion of the Theorem. For the remaining proof of the representation (.0, let µ be a measure on the interval [0, d] with first moments given by m n and let ν denote the measure on the interval [0, 1] obtained from µ by the linear transformation x x/d. We write p (µ for p (m n and z (µ for z (m n. Invariance of the canonical moments under linear transformations yields p (µ = p (ν. The recursion variables of the measure ν can be decomposed as (.1 z (ν = (1 p 1 (νp (ν. The Stieltjes-transform of µ has a continued fraction expansion (. d 0 dµ(x ζ x = 1 ζ z 1 (µ z 1 (µz (µ ζ (z (µ + z 3 (µ z 3 (µz 4 (µ ζ (z 4 (µ + z 5 (µ... for ζ C \ [0, d]. Since the measure µ is obtained from ν by linear transformation, the continued fraction expansion of the Stieltjes-transform of µ can be written in terms of the recursion coefficients of ν, d 0 dµ(x ζ x = 1 ζ dz 1 (ν d z 1 (νz (ν ζ d(z (ν + z 3 (ν d z 3 (νz 4 (ν ζ d(z 4 (ν + z 5 (ν... [see Theorem in Dette and Studden (1997]. A continued fraction expansion as in (. is unique, which yields dz (ν = z (µ. With equation (.1 we obtain p (µ = p (ν = z (ν 1 p 1 (ν = 1 z (µ d 1 p 1 (µ for > 1. The first canonical moment is given by p 1 (m n = m 1 m 1 m + 1 m 1 = m 1 d = z 1(m n. d and equation (.0 follows by an induction argument. The following Theorem is essential for the asymptotic investigations in Section 4 and gives the distribution of the the vector z n = (z 1,..., z n T corresponding to a random vector point on the moment space M n ([0,. Theorem.3. Let m n M n ([0, be governed by a law with density g n (γ,δ, then the recursion variables z n = ψ n (m n defined by (.7 are independent and gamma distributed, that is z Gamma(γ + n + 1, δ = 1,..., n. 8

9 Proof: By its definition in (.7, the random variable z depends only on the moment m 1,..., m, therefore the Jacobi matrix of the mapping ψ n is a lower triangular matrix. We obtain for the Jacobian determinant m n z n n = m z n = n (m 1 m 1 = z 1... z 1 = = z n, where the third identity follows from the definition of the z i in (.7. Considering the second representation of the density in (.19, this yields the claimed distribution. We will show in Section 3 that densities of the form (.16 arise naturally as distributions of moments corresponding to random spectral measures. We conclude this section with a discussion of distributions on the moments space corresponding to measures on R. For the sae of brevity we restrict ourselves to moment spaces of odd dimension, that is M n 1 (R = { m n 1 (µ = (m 1 (µ,..., m n 1 (µ T µ P(R }. To derive a class of distributions on M n 1 (R we consider the moment space M n 1 ([ s, s] with s and a density of the form (.16 with parameters varying with s. The proof of the following result is similar to the proof of Theorem. and therefore omitted. Theorem.4. Denote by f (γ(s,δ (s n 1 where the parameters satisfy the density defined in (.16 on the moment space M n 1 ([ s, s], γ (s 1 = δ 1 s + o(1, δ s 1 = δ 1 s + o(1, γ (s = γ + o(1, δ (s = δ s + o(s with γ > 1, δ > 0. Then f (γ(s,δ (s n 1 converges point-wise to the function h (γ,δ δ 1 (.3 n 1(m n 1 = π exp ( δ 1 b (m n 1 n 1 δ γ +n Γ(γ + n aγ (m n 1 exp ( δ a (m n 1 1 {a (m n 1 >0}. Moreover, the function h (γ,δ n 1 defines a density on the moment space M n 1 (R. The following result is the analogue of Theorem.3. Theorem.5. Let m n 1 M n 1 (R be a random vector with density h (γ,δ n 1 defined in (.3. Then the random recursion coefficients (b 1, a 1..., a n 1, b n T = ξ n 1 (m n 1 in the recurrence relation (.3 for the orthogonal polynomials associated with m n 1 by (.4 are independent and b N (0, 1 δ 1, a Gamma(γ + n, δ. 9

10 Proof: Suppose µ is a measure with first moments given by m n 1 and let P 1 (x,..., P n (x denote the corresponding monic polynomials. The recursion coefficients can be calculated by (.1 and (.13. This implies that the coefficient b depends only on the moments m 1,..., m 1, while a depends only on m 1,..., m. Consequently, the Jacobi matrix of ξ n 1 is a lower triangular matrix. From (.1 we obtain m a = a and by the identity (.13 it follows m 1 b = b The Jacobian determinant is therefore given by m n 1 (n 1 ξ n 1 (m n 1 = m 1 b (n 1 = R R x P (xdµ(x = a 1... a 1 x P 1 (xdµ(x = a 1... a 1. m a mn 1 b n n 1 (a 1... a 1 (a 1... a 1 = = a n 1. Considering the product structure in the density h (γ,δ n 1 in terms of the recursion coefficients, the assertion of the theorem follows. 3 Random spectral measures Let A be a linear, self-adjoint operator on a n-dimensional Hilbert space H with inner product, and cyclic vector e 1 H (i.e., e 1, Ae 1,..., A n 1 e 1 are linearly independent. The spectral Theorem yields the existence of a unique probability measure µ on the real Borel field, such that (3.1 e 1, A n e 1 = m n (µ for all n 1 [see Dunford and Schwartz (1963]. This measure defines the unitarily equivalent L -space in which the operator A is represented by the multiplication f(x xf(x. We call µ spectral measure of the matrix A. If λ 1,..., λ n denote the eigenvalues of A and u 1,..., u n is a corresponding system of orthonormal eigenvectors, the spectral measure can be written as (3. µ = n w i δ λi i=1 where the weights are given by w i = u i, e 1 and δ x denotes the Dirac measure in the point x. The identity (3. follows easily from the fact that the matrix with columns u 1,..., u n 10

11 diagonalizes A. If the eigenvalues are distinct, lie in the interior of T {[a, b], [0,, R} and the vector of weights (w 1,..., w n T is contained in the simplex Sim n = { (w 1,..., w n T R n wi > 0, n i=1 } w i = 1 then the first n 1 moments of the spectral measure satisfy m n 1 (µ Int M n 1 (T which follows by an application of Theorem in Dette and Studden (1997. We consider in this section random spectral measures associated with three central distributions in random matrix theory: the Hermite (or Gaussian, the Laguerre and the Jacobi ensemble. These classical ensembles are distributions on the space of self-adjoint matrices A K n n with real (K = R, complex (K = C or quaternion entries (K = H. The Hermite ensemble arises in physics and is the distribution of the matrix A = 1 (X + X, where all real entries in X K n n are independent and standardnormal distributed and X denotes the adjoint of X. The eigenvalues of the Hermite ensemble have the joint density (3.3 f G (λ = c G (λ β n e λ i / i=1 where β = dim R K, (λ = i<j (λ j λ i is the Vandermonde determinant and c G is a normalizing constant [see Mehta (004]. The Laguerre ensemble appears in the study of singular values of a Gaussian matrix, it has the eigenvalue density (3.4 f L (λ = c a L (λ β n i=1 λ a i e λ i 1 {λi >0}, with parameter a > 1 and a normalizing constant c a L. Finally, the Jacobi ensemble is motivated by multivariate analysis of variance [MANOVA; see Muirhead (198] and is defined by the eigenvalue density (3.5 f J (λ = c a,b J (λ β λ a i (1 λ i b 1 {0<λi <1}. i=1 with a, b > 1 and a normalizing constant c a,b J. It is a common feature of all three classical ensembles that the matrix of the orthonormal eigenvectors is Haar distributed on the group of orthogonal (unitary/symplectic matrices and independent from the eigenvalues [see Dawid (1977]. As a consequence, the matrix distribution is uniquely determined by the eigenvalue density. Since the eigenvector matrix is Haar distributed, the first row of the matrix is Haar 11

12 distributed on the unit sphere in K n and the weights w 1,..., w n follow a Dirichlet distribution with density (3.6 Γ( nβ Γ( β wβ/ 1 n 1... w β/ 1 n 1 {w Simn}. The identity for the spectral measure in (3. motivates the following definition of distributions on the set of probability measures. Definition 3.1. Let µ = w 1 δ λ1 + + w n δ λn denote a probability measure with random support points λ = (λ 1,..., λ n and random weights with w 1,..., w n with Dirichlet density (3.6, where the weights and (random support points are independent. (i If the density of λ is given by (3.3, we call µ random spectral measure of the Gaussian ensemble, i.e. µ GβE n. (ii If the density of λ is given by (3.4, we call µ random spectral measure of the Laguerre ensemble, i.e. µ LβE n (a. (iii If the density of λ is given by (3.5, we call µ random spectral measure of the Jacobi ensemble, i.e. µ JβE n (a, b. Note that these distributions are well defined for all β > 0. If β {1,, 4}, we obtain the spectral measures of matrices from the classical ensembles. The results in this section show that there is a connection between classical ensembles and the distributions on moment spaces as given in Section. More precisely, the moments (3.1 of the spectral measures are distributed according to the densities defined on M n ([0, 1] and obtained on the unbounded moment spaces by Theorem. and Theorem.4. Our first Theorem considers the spectral measure of the Jacobi ensemble. The proof relies on a tridiagonal matrix model for the JβE n (a, b distribution for all values of β > 0, which was proved in a recent paper by Killip and Nenciu (004. Theorem 3.. Let µ JβE n (γ 0, δ 0 denote a random spectral measure of the Jacobi ensemble, then the distribution of the corresponding random moment vector m n 1 (µ on M n 1 ([0, 1] is absolute continuous with density f (γ,δ n 1 defined in (.16 where the parameters of the density are given by for 1 n and for 1 n 1. γ 1 = ( β (n + γ 0 ; δ 1 = ( β (n + δ 0 γ = ( β (n ; δ = ( β (n 1 + γ 0 + δ 0 1

13 Proof: It follows from Theorem. and Proposition 5.3 in Killip and Nenciu (004 that µ is the spectral measure of the tridiagonal matrix d 1 c 1. J n = c 1 d cn 1, c n 1 d n where the entries are given by d =p (1 p 3 + p 1 (1 p c = p 1 (1 p p (1 p 1 with p 1 = p 0 = 0 and p 1,..., p n 1 are independent random variables distributed as { ( Beta n β, n β + γ δ 0 +, even, p Beta ( n 1β + γ , n 1 β + δ , odd. Note that Killip and Nenciu (004 define the Jacobi ensemble by the eigenvalue density c (λ β n i=1 ( λ i γ 0 ( + λ i δ 0 1 { <λi <} and therefore the matrix found in their paper is the transformed matrix 4J n I n. Additionally, they wor with a beta distribution on the interval [ 1, 1] which is obtained from the usual beta distribution on [0, 1] by the transformation x 1 x. The tridiagonal matrix J n defines monic polynomials P 1 (x,..., P n (x via a recursion (.3 with recursion coefficients b = d (1 n, a = c (1 n 1. Indeed, the polynomial P (x is the characteristic polynomial of the upper left ( -subbloc of the matrix J n. The orthogonality measure of these polynomials is precisely the spectral measure µ [see Deift (000 for the corresponding statement for orthonormal polynomials]. Therefore, the recursion coefficients in the recursion (.3 of the monic polynomials orthogonal with respect to the measure µ are given by b =p (1 p 3 + p 1 (1 p ; = 1,..., n, a =p 1 (1 p p (1 p 1 ; = 1,..., n 1. 13

14 By identity (.5 and (.6, p n 1 = (p 1,..., p n 1 T is exactly the vector of canonical moments of the spectral measure µ and by definition, their joint density is given by f p (p n 1 =c =c n 1 n 1 p (n ( 1 1β/4+γ 0 1 (1 p 1 (n ( 1 1β/4+δ 0 p (n β/4 1 (1 p (n β/4+γ 0+δ 0 +1 p (n β/+γ 0 1 (1 p 1 (n β/+δ 0 p (n β/ 1 (1 p (n 1β/+γ 0+δ Since the eigenvalues of the matrix J n are contained in the interval (0, 1, the moments m n 1 (µ = ϕn 1(p n 1 of the spectral measure are in the interior of the moment space M n 1 ([0, 1]. The Jacobian of the transformation ϕ 1 n 1 is given by n (p (1 p n 1, which gives for the density of the random moments m n 1 (µ f m (m n 1 =c =c p 1 (m n 1 (n β/ (n +γ 0 (1 p 1 (m n 1 (n β/ (n +δ 0 n 1 p (m n 1 (n β/ (n 1 1 (1 p (m n 1 (n 1β/ (n 1+γ 0+δ 0 +1 p 1 (m n 1 (β/ (n +γ 0 (1 p 1 (m n 1 (β/ (n +δ 0 n 1 p (m n 1 (β/ (n (1 p (m n 1 (β/ (n 1+γ 0+δ 0. This is a density as in (.16 with the asserted parameters. An interesting case is obtained by the choice β = 4. Here the parameters γ, δ of the density f (γ,δ n 1 do not depend on. If additionally γ 0 = δ 0 = 0, the vector of moments m n 1 is uniformly distributed on M n 1 ([0, 1]. In other words, starting from a moment vector m n 1 drawn uniformly from the moment space, we obtain a random measure µ = n w δ λ with first moments given by m n 1 and with support points distributed according to the density (3.7 f λ (λ = (λ 4 1 {0 λ 1 }. 14

15 The roots of the n-th orthogonal polynomial P n (x with respect to the measure µ are precisely the support points λ 1,..., λ n. Consequently the roots of the n-th random orthogonal polynomial P n (x associated with m n 1. are distributed according to the density (3.7 [see also Bire and Dette (009]. The following results show that the spectral measures of the Laguerre ensemble and Hermite ensemble have moments distributed according to the density obtained in Theorem. and Theorem.4, respectively. The proofs are analogous to the proof of Theorem 3. and use the tridiagonal matrix models for the Laguerre and Hermite ensemble provided by Dumitriu and Edelman (00. Theorem 3.3. Let µ LβE n (γ 0 denote a random spectral measure of the Laguerre ensemble, then the distribution of the corresponding random moment vector m n 1 (µ on M n 1 ([0, is absolute continuous with density g n (γ,δ defined by (.19, where the parameters of the density are given by δ = 1 for 1 n 1 and γ 1 =( β (n + γ 0 1 n, γ =( β (n 1 n 1. Theorem 3.4. Suppose µ GβE n is a random spectral measure of the Gaussian ensemble, then the distribution of the corresponding random moment vector m n 1 (µ on the moment space M n 1 (R is absolute continuous with density h (γ,δ n 1 defined by (.3 where the parameters of the density are given by δ 1 = 1 (1 n, δ =1 (1 n 1, and γ = ( β (n 1 (1 n. 4 Wea convergence of random moments In this section we study the probabilistic properties of random vector on the moment spaces M n ([a, b], M n ([0, and M n (R distributed according to the measures introduced in Section. We begin with random moments defined on the moment space corresponding to probability measures on a compact interval. Chang et al. (1993 and Gamboa and Lozada-Chang (004 investigated the uniform distribution on M n ([a, b], and we first demonstrate that wea convergence of random moment vectors can be established for a rather broad class of distributions on M n ([a, b]. An important role in the discussion of moment spaces corresponding to probability measures with bounded support [a, b] plays the arcsine distribution ν with density dν(x = 1 π (x a(b x 1 {a<x<b}dx. 15

16 The canonical moments of the arcsine distribution are given by 1/ [see Dette and Studden (1997] and therefore its sequence of moments could be considered as the center of the moment space M([a, b]. The following statements establish the asymptotic properties of the (random canonical moments corresponding to distributions on the moment space M n ([a, b] defined in (.14. The only assumption necessary to obtain almost sure convergence is that for all ε > 0 (4.1 Throughout this paper the symbol 1 +ε f (xdx > 0. 1 ε D stands for wea convergence. Theorem 4.1. Suppose that the distribution of the random moment vector m n M n ([a, b] is absolute continuous with density f n defined in (.14, where the functions f satisfy for all ɛ > 0 condition (4.1, and denote by p (n the -th canonical moment of m n ( = 1,..., n. (a If n, then almost surely for any 1 p (n n (b If additionally the function f in the density (.14 is continous at 1 and f ( 1 > 0, then the -th canonical moment corresponding to m n satisfies (n 8n(p 1 D N (0, 1. Proof: For notational convenience, we consider only the case [a, b] = [0, 1]. We introduce the random variable q (n = p (n with density 1 n 1. c,n f (x + 1 ( 1 4 x n 1 { 1/<x<1/} and show q (n 0, which proves the first assertion of the Theorem. For 0 < ε < 1, let 4 U(ε = [ ε, ε], then clearly, 1 = c,n f (x + 1( 1 4 x n dx + c,n f (x + U(ε 1( 1 4 x n dx, C U(ε and we obtain 1 (c,n f (x + 1( 1 4 x n dx = 1 + f U(ε (x + 1( 1 4 x n dx U(ε C f U(ε C (x + 1( 1 4 x n dx U( ε f (x + 1( 1 4 x n dx f U(ε C (x + 1( 1 4 x n dx ( 1 4 ε 4 n U( ε f (x + 1 dx ( 1 4 ε n U(ε C f (x + 1 dx 16 ( 1 ε 1 4ε n c ε,f,

17 where c ε,f is a positive constant by condition (4.1 and independent of n. This yields P which implies ( ( q (n 1 ε U(ε C = c,n f (x + U(ε 1( 1 4 x n n dx c 1 1 4ε C ε,f, P n= By the Borel-Cantelli Lemma, q (n the Theorem. ( q (n U(ε C c 1 ε,f ( 1 ε n <. 1 4ε n= converges to 0 almost surely, which proves the first part of For a proof of part (b we note that the rescaled canonical moment 8n(p (n 1 has the density (4. c,n 1 8n (n f ( 1 + x 8n n (1 x 1 n { n<x< n}, and the assertion follows from the convergence of the density (4. to the density of the standard normal distribution [see the convergence Theorem by Scheffé (1947]. Clearly, f ( 1 + x 8n n (1 x 1 n { n<x< n} n f(1 / e x, while the factors in (4. not depending on x have the integral representation With the definitions I := we have the inequality ( 1 1 n c,n (n = f 8n ( 1 + n R f( 1 e x / dx, I n := n n x 8n n (1 x dx =: I n. n ( n f ( 1 1 x dx, n (4.3 I n I I n I n + I n I, and we need to show that the right hand side converges to 0. Obviously I n I converges to 0 by theorem of dominated convergence. For the first term on the right hand side of (4.3 we have for n sufficiently large n 1 I n I n f ( 1 f ( 1 + x 8n e x / dx = f ( 1 f ( 1 + x ne nx dx. n 1 17

18 For given ε > 0 we choose δ > 0, such that f ( 1 f ( 1 + x < ε for all x ( δ, δ then I n I n ε ne nx dx + f ( 1 f ( 1 + x ne nx dx ( δ,δ ( δ,δ C ε e x / dx + f ( 1 f ( 1 + x ne nx dx ( δ,δ C and the second integral converges to 0 by the theorem of monotone convergence. The wea convergence of the canonical moments was shown by Chang et al. (1993 for a uniform distribution on M n ([0, 1] (i.e., f 1. These authors showed wea convergence of the vector m (n of the first components of a uniformly distributed moment vector m n = (m 1,..., m n on the moment space M n ([0, 1], that is (4.4 8nA 1 (m (n m (ν D N (0, I, n where m (ν denotes the vector of the first moments of the arcsine distribution and A is a lower triangular matrix with entries ( i a i,j = i+ j i. i j By part (b of Theorem 4.1, it is easy to see that the wea convergence in (4.4 holds for the more general densities f n on M n ([0, 1]. We conclude this paper with a discussion of corresponding results for distributions on the non compact moment spaces M n ([0, and M n (R. In this case the analogs of the arcsine distribution in this context are the Marcheno-Pastur distribution defined by x(4 x dη(x = 1 {0<x<4} dx πx and Wigners semicircle distribution on the interval [, ], that is (4.5 dρ(x = 1 4 x 1 { <x<} dx π (see Nica and Speicher (006. The moments of the Marcheno-Pastur law η are the Catalan numbers c n defined by m n (η = c n = 1 ( n (4.6 n N, n + 1 n and the moments of the semicircle law ρ are given by { 1 m m n (ρ = m+1( if n = m m 0 if n = m 1. 18

19 Our next results establish the asymptotic properties of the quantities z corresponding to a random vector on the moment space M n ([0, with density g n (γ,δ defined in (.19. The following result is a well nown consequence of the asymptotic behavior of the density of the Gamma distribution and the proof therefore omitted. Theorem 4.. Suppose m n is a random vector of moments on the moment space M n ([0, with density g n (γ,δ, where the γ are fixed, δ 1 = = δ n = n, and let z (n denote the -th component of the vector z n = (z (n 1,..., z n (n. Then the standardized random variable z (n is asymptotically normal distributed, that is (n D n(z 1 N (0, 1. n By Theorem 4. the vector (n n(z 1 = n((z (n 1,..., z (n T (1,..., 1 T is asymptotically multivariate normal distributed. In order to derive a corresponding statement of the random vector m (n = ψ 1 (n (z we will use the Delta method and study first the image of the vector (1,..., 1 T under the mapping ψ 1 Lemma 4.3. Let (c n n 1 denote the sequence of Catalan numbers defined in (4.6, then. ψ n (c 1,..., c n = (1,..., 1 T. Proof: The proof presented here relies on the combinatorical interpretation of the Catalan numbers and a recursive algorithm given in Sibinsy (1968 to calculate the moments in terms of the variables z. The -th Catalan number counts the paths in N N starting in (0, 0 and ending in (, 0, where one is only allowed to mae steps in the direction (1, 1 or (1, 1. Sibinsy (1968 defines the triangular array {g i,j } i,j 0 by g i,j = 0 for i > j, g 0,j = 1 and the recursion (4.7 g i,j = g i,j 1 + z j i+1 g i 1,j, 1 i j. He showed that g, = m. Consequently, if z i = 1 (i = 1,,... the quantity g, is the number of paths through the lattice {(i, j} i,j 0, starting in (, and ending in (0, 0, where in each vertex we can only mae steps upward or to the left and where we are not allowed to cross the diagonal {(i, i}. This number is exactly the -th Catalan number c. Theorem 4.4. If the vector of random moments m n M n ([0, is governed by a law with density g n (γ,δ, where δ 1 = = δ n = n and the γ are fixed, then the projection m (n = Π n (m n of m n onto the first coordinates satisfies nc 1 (m (n D m (η N (0, I, n 19

20 where the vector m (η = (c 1,..., c T contains the first moments of the Marcheno-Pastur distribution and C is a lower triangular matrix with entries c 1,1 = = c, = 1, and ( ( i i c i,j = j < i. i j i j 1 Proof: It suffices to calculate the Jacobi matrix C = ψ 1 (z 0 z of the mapping ψ 1 at z 0 = (1,..., 1T, then the independence of the recursion variables z and Theorem 4. yield the convergence n(m (n m (η = nc(z (n z 0 D + o P (1 N (0, CC T. n Note that the moment m i depends only on z 1,..., z i and consequently C is a lower triangular matrix. To identify the entries of the matrix C we consider the triangular array {g i,j } i,j 0 defined in (4.7. For a fixed r with 1 r we introduce the notation u i,j = g i,j z r (z 0, and obtain a new triangular array {u i,j } i,j 0. Obviously we have u i,j = 0 for i > j and the other values of u i,j are determined by the initial condition u 0,j = 0 and the recursion (4.8 u i,j = u i,j 1 + u i 1,j + δ r,j i+1 g 0 i 1,j, 1 i j, were δ i,j denotes the Kronecer symbol and gi,j 0 is the coefficient in the recursion (4.8, if all z i are equal to 1, that is ( ( i + j i + j gi,j 0 =. i i 1 The numbers gi,j 0 are sometimes called generalized Catalan numbers [see Finucan (1976]. At the end of this proof we will show that the entries in the new triangular array are given by (4.9 u i,j = {( i+j i 1 ( i+j j r ( i+j i r 1 ( i+j i r 1 if j i r, if 0 j i < r. With this identity we obtain for the entries of the matrix C c i,r = m ( ( i i i (z 0 = u i,i = z r i r i r 1 for 1 r i, which proves the assertion of the Theorem. 0

21 The proof of (4.9 follows by an induction argument for the row number i. Obviously we have u 0,j = 0 and therefore u 1,j = 0 as long as j < r. The value of u 1,r is equal to g 0 0,r = 1 and for j > r we have u 1,j = 1. These values are also given by the formula (4.9. In the induction step i 1 i we perform an induction for j and have to distinguish between five cases with different recursion for u i,j. The scheme in (4.10 illustrates these cases in the i-th row. ( (i, i }{{} (1... (i, i (i, i + r }{{} (... (i, i + r 1 }{{} (3... (i, i + r }{{} (4... (i, i + r }{{} (5 (1 j = i: First consider r = 1, Then case (1 is identical to case (3 and u i,i =u i 1,i + gi 1,i 0 = ( ( i 1 i i 1 ( i 3 + i 1 ( i 1 i 1 i = ( ( i i 1 i 1 ( i i 1 ( i 3 = i ( i 1 i i. For r > 1 it is u i,i = u i 1,i = ( ( i 1 i r i 1 ( i r = i ( i r i 1 ( i r 1 i 1 ( i r = i ( i r i i r 1. ( i < j < i + r 1: u i,j = u i,j 1 + u i 1,j = ( ( i+j 1 j 1 r i+j 1 ( i r 1 + i+j 1 ( j r i+j 1 ( i r = i+j ( j r i+j i r 1 (3 j = i + r 1: The case r = 1 was considered in (1, for r > 1 we have u i,j = u i,j 1 + u i 1,j + gi 1,j 0 = ( ( i+j 1 j 1 r i+j 1 ( i r 1 + i+j 1 ( i i+j 1 ( i r + i+j 1 ( i 1 i+j 1 i = ( ( i+j 1 j 1 r i+j ( i r 1 + i+j 1 ( j r = i+j ( j r i+j (4 j = i + r: u i,j =u i,j 1 + u i 1,j = ( ( i+j 1 j 1 r i+j 1 ( i r 1 + i+j 1 i = ( ( i+j j r 1 i+j ( i r 1 = i+j ( i 1 i+j i r 1 (5 j > i + r: u i,j = u i,j 1 + u i 1,j = ( i+j 1 i 1 ( i+j 1 i r ( i+j 1 ( + i+j 1 i r 1 i i r 1. ( = i+j 1 ( + i+j 1 ( i+j ( i+j 1 i r j r 1 i 1 j r ( = i+j ( i+j i r 1 i r 1 We conclude the discussion of random moments considering the moment space M n 1 (R. Recall the bijective mapping (.11 from the interior of the moment space M n 1 (R onto 1

22 (R R + n 1 R corresponding to the range for coefficients in the recursive relation of the orthogonal polynomials (.3. The following results give the wea asymptotics of random recursion coefficients and moments and correspond to Theorem 4. and 4.4. The proof of Theorem 4.6 follows by similar arguments as presented in the proof of Theorem 4.4 and is therefore omitted. Theorem 4.5. Let the random vector m n 1 M n 1 (R be governed by a law with density h (γ,δ n 1 where γ > 1 is fixed and δ = n ( = 1,..., n. For fixed denote by b (n and by a (n the ( 1-th component of the vector ξ n 1 (m n 1 and the -th component, respectively. Then (n nb N (0, 1, n(a (n 1 D N (0, 1. n Theorem 4.6. Let the vector of random moments m n 1 M n 1 (R be governed by a law with density h (γ,δ n 1 where δ = n ( = 1,..., n. For N denote by m (n = Π n (m n 1 the projection onto the first coordinates and by m (ρ = Π (0, c 1, 0, c,... the vector of the first moments of the semicircle law defined in (4.5. Then nd 1 (m (n D m (ρ N (0, I, n where D is a lower triangular matrix with d i,j = 0 if i + j is odd and the remaining entries are given by ( ( i i d i,j = i j i j 1. By the results in Section 3, the moment density of the three classical ensembles is the moment density investigated asymptotically in this Section. Although for the random matrix ensembles the parameters γ, δ depend on n, only minor changes are necessary to obtain a wea convergence result for the first moments. In the case of the Jacobi ensemble, the canonical moment p (n follows a Beta distribution with parameters behaving lie β n and therefore, (n 4βn(p 1 D N (0, 1 n and we obtain easily the right scaling for the ordinary moments in the following Corollary. Corollary 4.7. Let µ n JβE n (γ 0, δ 0 be a spectral measure of the Jacobi ensemble, then the first moments m (µ n of µ n satisfy 4βnA 1 D (m (µ n m (ν N (0, I, n where m (ν is the moment vector of the arcsine measure and A is the matrix in (4.4.

23 In particular, the moment convergence implies the wea convergence of the spectral measure to the arcsine measure. This is also a consequence of the well-nown convergence of the emprical eigenvalue distribution to the arcsine measure, since the (unscaled moments of the spectral measure have the same asymptotic behaviour as the moments of the empirical eigenvalue distribution. By Corollary 4.7, the fluctuations around this limit in terms of the moments are Gaussian. A corresponding result holds for the Laguerre ensemble with rescaled eigenvalue density f L (λ = c γ 0 L (λ β i=1 λ γ 0 i e βnλ i/ 1 {λi >0}. For the first moments of a spectral measure µ n with this eigenvalue density we obtain from Theorem 3.3 together with Theorem.3 the asymptotic βn/c 1 (m (µ n m (η D n N (0, I. From Theorem 3.4 and Theorem.5 we can deduce for a spectral measure µ n of the rescaled Gaussian Ensemble with eigenvalue density f G (λ = c G (λ β n i=1 e βnλ i /4. the wea convergence of the first moments βn/d 1 D (m (µ n m (ρ N (0, I. n The different scaling in these two cases comes from the scaling necessary to obtain wea convergence of the recursion parameters z and the recursion coefficients a, b, respectively. Again, this convergence results can be related to the convergence of the empirical eigenvalue density to the Marcheno-Pastur law and the semicircle distribution and give the fluctuations around the limit distribution. Acnowledgements. The authors are grateful to Martina Stein who typed parts of this paper with considerable technical expertise. The wor of the authors was supported by the Sonderforschungsbereich Tr/1 (project C, Fluctuations and universality of invariant random matrix ensembles and in part by a grant of the Deutsche Forschungsgemeinschaft De 50/-3. References Bire, M. and Dette, H. (009. A note on some random orthogonal polynomials on a compact interval. Proceedings of the American Mathematical Society., 137 (10:

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