A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.


Introduction
For Riemannian symmetric spaces G/K of the compact or non-compact type, there is a well-known contraction principle which states that under suitable scaling, the spherical functions ϕ λ of G/K tend to the spherical functions ψ λ of the tangent space of G/K in the base point, which is a symmetric space of the flat type: lim n→∞ ϕ nλ (exp(x/n)) = ψ λ (x).
See [C] and, for a more recent account, [BO]. This curvature limit, also known as Mehler-Heine formula, extends to the more general setting of hypergeometric functions associated with root systems, which converge under rescaling to generalized Bessel functions. This is proven, by a limit transition in the Cherednik operators, in [dJ]; see also [BO] for a different approach. In the compact rank one case, the contraction principle is a weak version of the classical Hilb formula for Jacobi polynomials (see Theorem 8.21.12 of [Sz]), which provides in addition a precise estimate on the rate of convergence.
In this paper, we prove in Theorem 2.4 a Mehler-Heine formula with a precise estimate on the error term for a certain class of orthogonal polynomials associated with root systems, which in particular encompasses the spherical functions of compact Grassmannians. This result is a "compact" analogue of the Theorem 5.4 in [RV], which gives a scaling limit with error bounds for hypergeometric functions in the dual, non-compact setting. In the second part of the paper, we shall use the Mehler-Heine formula 2.4 in order to establish a central limit theorem for random walks on the semi-lattice of dominant weights parametrizing the unitary dual of a compact Grassmannian.
To become more precise, we consider in this paper the compact Grassmannians G p,q (F) = G/K over one of the (skew-) fields F = R, C, H, with G = SU (p + q, F) and K = S(U (q, F) × U (p, F)), where p ≥ q ≥ 1. Via polar decomposition of G, the double coset space G//K = {KgK : g ∈ G} may be topologically identified with the fundamental alcove A 0 := {x = (x 1 , . . . , x q ) ∈ R q : π 2 ≥ x 1 ≥ x 2 ≥ · · · ≥ x q ≥ 0}, with x ∈ A 0 being identified with the matrix (1.1) Here we use the diagonal matrix notation x = diag(x 1 , . . . , x q ), and the functions sin, cos are understood component-wise. For details, see Theorem 4.1 in [RR]. The spherical functions of G p,q (F) can be viewed as Heckman-Opdam polynomials of type BC q , which are also known as multivariable Jacobi polynomials. They may be described as follows: denote by F BC (λ, k; ·) the Heckman-Opdam hypergeometric function associated with the root system with spectral variable λ ∈ C q and multiplicity parameter k = (k 1 , k 2 , k 3 ) ∈ R 3 corresponding to the roots ±2e i , ±4e i and 2(±e i ± e j ). Fix the positive subsystem and the associated semi-lattice of dominant weights, P + = {λ ∈ (2Z) q : λ 1 ≥ λ 2 ≥ · · · ≥ λ q ≥ 0}. (1.2) Then the set of spherical functions of G p,q (F) is parametrized by P + and given by ϕ p λ (a x ) = F BC (λ + ρ p , k(p), ix) =: R p λ (x), λ ∈ P + (1.3) with multiplicity parameter k(p) = (d(p − q)/2, (d − 1)/2, d/2), where d = dim R F ∈ {1, 2, 4} and (1.5) The functions R p λ are the Heckman-Opdam polynomials associated with the root system R (called Jacobi polynomials in the following) and with multiplicity k(p), normalized according to R p λ (0) = 1. We refer to [H], [HS], [O2] for Heckman-Opdam theory in general, and to [RR] and the references cited there for the connection with spherical functions in the compact BC case. Notice that our notion of F BC coincides with that of Heckman,Opdam and [R2], [RV], while it differs from the geometric notion in [RR]. Theorem 4.3 of [RR] corresponds to (1.3).
In Theorem 4.2 of [RR], the product formula for spherical functions of (G, K) was written as a formula on A 0 which could be analytically extended to a product formula for the Jacobi polynomials R p λ with multiplicity k(p) corresponding to arbitrary real parameters p > 2q − 1. This led to three continuous series of positive product formulas for Jacobi polynomials corresponding to F = R, C, H and to associated commutative hypergroup structures on A 0 ; see [J] and [BH] for the notion of hypergroups. Using a Harish-Chandra-type integral representation for the R p λ , we shall derive the Mehler-Heine formula 2.4 with a precise asymptotic estimate for the Jacobi polynomials R p λ in terms of Bessel functions associated with root system B q on the Weyl chamber This Mehler-Heine formula will be the key ingredient for the main result of the present paper, a central limit theorem for random walks on the semi-lattice P + , which parametrizes the spherical unitary dual of G/K. To explain this CLT, let us first recall that via the GNS representation, the spherical functions ϕ λ , λ ∈ P + of (G, K), which are necessarily positive definite, are in a one-to-onecorrespondence with the (equivalence classes of) spherical representations (π λ , H λ ) of G, that is those irreducible unitary representations of G whose restriction to K contains the trivial representation with multiplicity one, see [F] or Chap. IV of [Hel]. The decomposition of tensor products of spherical representations into their irreducible components leads to a probability preserving convolution * d,p and finally a hermitian hypergroup structure on the discrete set P + ; see [Du] and [K3]. Following e.g. [BH], [Z], and [V1], we introduce random walks (S d,p n ) n≥0 on P + associated with * d,p and derive some limit theorems for n → ∞. The main result of this paper will be the CLT 3.15. This CLT implies the following result for G p,q (F) = G/K.
1.1 Theorem. Let (π λ , H λ ) be a non-trivial spherical representation of G associated with λ ∈ P + \{0}. Let u λ ∈ H λ be K-invariant with u λ 2 = 1. For each n ∈ N, decompose the n-fold tensor power (π ⊗,n λ , H ⊗,n λ ) into its finitely many irreducible unitary components where the components are counted with multiplicities. Consider the orthogonal projections p τn : H ⊗,n λ → H τn and a P + -valued random variable X n,λ with the finitely supported distribution τn p τn (u ⊗,n λ ) 2 2 δ τn ∈ M 1 (P + ) with the point measures δ τn at τ n . Then, for n → ∞, which is the distribution of a Laguerre ensemble on C. The modified variance parameter m(λ) > 0 is quadratic in λ and given explicitly in Lemma 3.6 below.
For q = 1, the CLT 3.15 has a long history as a CLT for random walks on Z + whose transition probabilities are related to product linearizations of Jacobi polynomials. This includes random walks on the duals of SU (2) and (SO(n), SO(n − 1)) in [ER] and [G]. See also [V1] for further onedimensional cases. For q ≥ 2 our results are very closely related to the work [CR] of Clerc and Roynette on duals of compact symmetric spaces. For a survey on limits for spherical functions and CLTs in the non-compact case for q = 1 we refer to [V3].

A Mehler-Heine formula
In this section we derive a Mehler-Heine formula for the Jacobi polynomials R p λ (λ ∈ P + ), describing the approximation of these polynomials in terms of Bessel functions with a precise error bound. Our result will be based on Laplace-type integrals for the Jacobi polynomials and the associated Bessel functions, where we treat the group cases with integers p ≥ q as well as the case p ∈ R with p ≥ 2q − 1 beyond the group case. The integral representation for R p λ below is a special case of a more general Harish-Chandra integral representation for hypergeometric functions F BC in [RV]. To start with, let us introduce some notation: Let H q x * := x t = x} denote the space of Hermitian matrices over F, and denote by ∆(x) the determinant of x ∈ H q (F), which may be defined as the product of (right) eigenvalues of x. We mention that for F = H, this is just the Moore determinant, which coincides with the Dieudonné determinant if x is positive semi-definite, see e.g [A]. On H q (F), we consider the power functions with the principal minors ∆ r (a) = det((a ij ) 1≤i,j≤r ) of the matrix a = (a ij ) 1≤i,j≤q ∈ H q (F), see [FK]. We introduce the matrix ball Here dw is the Lebesgue measure on the ball B q , According to Theorem 2.4 of [RV], the Heckman-Opdam hypergeometric function F BC (λ, k(p), x) with λ ∈ C q , x ∈ R q and k(p) as in (1.4) has the following integral representation for p ∈ R with p > 2q − 1: where U 0 (q, F) denotes the identity component of U (q, F) and It is easily checked that U 0 (q, F) may be replaced by U (q, F) in the domain of integration. Notice further that x → g x (u, w) extends to a holomorphic function on C q . As the principal minors ∆ r (a) are polynomial in the entries of a ∈ H q (F), it follows that x → ∆ λ/2 (g x (u, w)) extends to a holomorphic function on C q for each λ ∈ P + . In view of relation (1.3), this leads to the following integral representation for the Jacobi polynomials R p λ : 2.1 Proposition. Let p ∈ R with p > 2q − 1 and k(p) = (d(p − q)/2, (d − 1)/2, d/2) with d ∈ {1, 2, 4}. Then the Jacobi polynomials R p λ , λ ∈ P + , have the integral representation We next turn to the Bessel functions which will show up in the Mehler-Heine formula. They are given in terms of Bessel functions of Dunkl type which generalize the spherical functions of Cartan motion groups; see [dJ] and [O1] for a general background. We denote by J B k the Bessel function which is associated with the rational Dunkl operators for the root system B q = {±e i , ±e i ±e j : 1 ≤ i < j ≤ q} and multiplicity k = (k 1 , k 2 ) corresponding to the roots ±e i and ±e i ± e j . We shall be concerned with multiplicities which are connected as follows to the BC q multiplicities k(p) from (1.4): For such k on B q , we use the notion It is well-known that for integers p ≥ q, the ϕ p λ are the spherical functions of the Euclidean symmetric is the Cartan motion group associated with the Grassmannian G p,q (F). Hereby the double coset space G 0 //K is identified with the Weyl chamber C such that x ∈ C corresponds to the double coset of (I p × I q , (I p−q , x) ) ∈ G 0 , and in this way, K-biinvariant functions on G 0 may be considered as functions on C. Two functions ϕ p λ and ϕ p µ coincide if and only if λ and µ are in the same Weyl group orbit. Finally, the bounded spherical functions are exactly those ϕ p λ with λ ∈ C. The Bessel functions ϕ p λ with d = dim F R and not necessarily integral parameter p are closely related to Bessel functions on the symmetric cone of positive definite q × q-matrices over F, see Section 4 of [R1]. It has been shown there that for p > 2q − 1, they have a positive product formula which generalizes the product formula in the Cartan motion group cases and leads to a commutative hypergroup structure on the Weyl chamber C.
2.3 Remark. There are finitely many geometric cases which are not covered by the range p ∈ ]2q − 1, ∞[, namely the indices p ∈ {q, q + 1, . . . , 2q − 1}. In these cases, the Jacobi polynomials R p λ and the Bessel functions ϕ p λ both admit interpretations as spherical functions and have an integral representation similar to that above, by the following reasoning: According to Lemma 2.1 of [R2], the measure m p ∈ M 1 (B q ) with p ∈ N, p ≥ 2q is just the pushforward measure of the normalized Haar For p ∈ {q, q + 1, . . . , 2q − 1}, we now define the measure m p ∈ M 1 (B q ) in the same way as a pushforward measure of the Haar measure on U (p, F). (But in contrast to the case p ≥ 2q, it will not have a Lebesgue density in these cases). From the integral representations (3.3) and (4.4) of [R1] for the Bessel functions, as well as Theorem 2.1 of [RV] and relation (1.3) between Jacobi polynomials and hypergeometric functions, one obtains that the integral representations of Proposition 2.1 and Lemma 2.2 extend to the case p ∈ {q, q + 1, . . . , 2p − 1}.
We shall now compare the integral representations of Proposition 2.1 and Lemma 2.2, which will lead to the following quantitative Mehler-Heine (or Hilb-type) formula.

Thus in particular,
Notice that the estimate of Theorem 2.4 is uniform in p, a fact which was to our knowledge so far not even noticed in the rank-one case. We conjecture that the statement of this theorem remains correct for p ∈ [q, ∞[.

Proof.
We only consider the case p > 2q − 1 where the proof is based on Proposition 2.1 and Lemma 2.2. By the previous remark, the cases p = q, q + 1, . . . , 2q − 1 can be treated in the same way. Notice that it suffices to check uniformity in p for p > 2q − 1. We substitute w → u * w * in the integral (2.6) and obtain Denoting the trace of the upper left r × r-block of a q × q-matrix by tr r , we have Furthermore, by Proposition 2.1, Telescope summation yields the well-known estimate We thus obtain We now investigate ∆ r (g ix (u, w)) (λr −λr+1)/2 more closely. As x, u, w run through compacta, we obtain that uniformly in x, u, w, Using the power series for ln(1 + z), we further have Notice that y := u −1 (ixw + w * ix)u is skew-Hermitian, that is y * = −y. Therefore tr r (y) = −tr r (y), which implies that Re(tr r (y)) = 0. It follows that Note that these considerations apply for all fields F = R, C, H. It follows that there exists a constant From this inequality we obtain by the mean value theorem that for all x ∈ A 0 and λ ∈ P + , These estimates together with (2.8) imply the assertion.
2.5 Example (The rank one case). For q = 1 the Jacobi polynomials R p λ associated with root system BC 1 = {±e 1 , ±2e 1 } are classical one-variable Jacobi polynomials in trigonometric parametrization. For integers p, the associated Grassmannians are the projective spaces P p (F). For the details, recall that the classical Jacobi polynomials R (α,β) n with the normalization R (α,β) n (1) = 1 are given by see also Section 5 of [RR]. In the rank one case, the U (1, F) integral in representation (2.5) cancels by invariance of ∆ under unitary conjugation. Thus (2.5) reduces to (cos x + it sin x) 2n (1 − t 2 ) (p−3)/2 dt.
If F = C, then d = 2 and B 1 = {z ∈ C : |z| ≤ 1}. Using polar coordinates z = te iθ , one obtains The quaternionic case can be treated in a similar way. These formulas are just special cases of a wellknown Laplace-type integral representation for Jacobi polynomials with general indices α ≥ β ≥ −1/2; see e.g. Section 18.10 of [OLBC]. Let us finally mention that the Mehler-Heine formula 2.4 corresponds to Theorem 8.21.12 of [Sz] and that in the case of rank two (q = 2), the Jacobi polynomials of type BC were first studied by Koornwinder [K1], [K2].
3 Random walks on the dual of a compact Grassmannian and on P + Recall that for integers p ≥ q the functions ϕ λ := ϕ p λ , λ ∈ P + form the spherical functions of the compact Grassmannians G/K = G p,q (F). As functions on G, they are positive-definite. In other words, the Jacobi polynomials (R p λ ) λ∈P+ are just the hypergroup characters of the compact double coset hypergroups G//K ∼ = A 0 . We now recapitulate the associated dual hypergroup structures on P + .
3.1 Dual hypergroup structures on P + . As mentioned in the introduction, there is a one-to-one correspondence between the set of (positive definite) spherical functions of G/K, which is parametrized by P + , and the spherical unitary dual of G/K, i.e. the set G K of all equivalence classes of irreducible unitary representations (π, H) of G whose restriction to K contains the trivial representation with multiplicity one. Here a representation (π, H) ∈ G K and its spherical function ϕ π are related by ϕ π (x) = u, π(x)u for x ∈ G with some K-invariant vector u ∈ H with u 2 = 1, which is determined uniquely up to a complex constant of absolute value 1. Now consider λ, µ ∈ P + with associated spherical functions ϕ λ , ϕ µ and the corresponding representations (π λ , H λ ), (π µ , H µ ) ∈ G K with K-invariant vectors u λ , u µ . The tensor product (π λ ⊗ π µ , H λ ⊗ H µ ) is a finite dimensional unitary representation of G which decomposes into a finite orthogonal sum of irreducible unitary representations (τ k , H k ) where some of them may appear several times. Consider the orthogonal projections p k : with k p k (u λ ⊗ u µ ) 2 2 = 1. For λ, µ, τ ∈ P + we now define c λ,µ,τ ≥ 0 as p k (u λ ⊗ u µ ) 2 2 , whenever (τ k , H k ) = (π τ , H τ ) appears above with a positive part, and 0 otherwise. For λ, µ ∈ P + we define the probability measure δ λ * d,p δ µ := τ ∈P+ c λ,µ,τ δ τ ∈ M 1 (P + ) (3.1) with finite support. By its very construction, this convolution can be extended uniquely in a weakly continuous, bilinear way to a probability preserving, commutative, and associative convolution on the Banach space M b (P + ) of all bounded, signed measures on P + . Moreover, as all spherical functions are R-valued in our specific examples, the contragredient representation of any element in G K is the same representation, i.e., the canonical involution . * on P + ∼ = G K is the identity. In summary, (M b (P + ), * d,p ) is a commutative Banach- * -algebra with the complex conjugation µ * (A) := µ(A) as involution. Moreover, (P + , * d,p ) becomes a so-called hermitian hypergroup in the sense of Dunkl, Jewett and Spector; see [Du], [J], [BH]. The Haar measure on this hypergroup, which is unique up to a multiplicative constant, is the positive measure ω = λ∈P+ h(λ)δ λ with where the first two equations follow from general hypergroup theory (see [J]) and the last one from the theory of Gelfand pairs (see e.g. [F]). The coefficients c λ,µ,τ of the convolution * d,p on P + are related to the unique product linearization of the Jacobi polynomials R p λ . It is clear by our construction that for integers p ≥ q, all c λ,µ,τ are nonnegative with τ ∈P+ c λ,µ,τ = 1.
3.2 Random walks on P + . As before, we fix d = 1, 2, 4 and p ∈ [q, ∞[. We also fix an admissible probability measure ν = µ p µ δ µ ∈ M 1 p (P + ), and consider a time-homogeneous Markov chain (S d,p n ) n≥0 in discrete time on P + starting at time 0 in the origin 0 ∈ P + and with transition probability Such Markov-chains are called random walks on (P + , * d,p ) associated with ν. It is well-known and can be easily checked by induction that for all n the n-fold convolution power ν (n) := ν * d,p . . . * d,p ν ∈ M 1 (P + ) exists, and that ν (n) is just the distribution P S d,p n of S d,p n .
In view of the central limit theorem 1.1, we give an interpretation of these convolution powers δ (n) λ for integers p ≥ q and λ ∈ P + with λ = 0 in terms of representation theory. We expect that this result is well-known, but we do not know an explicit proof in the literature.
3.3 Lemma. Let (π λ , H λ ) be the non-trivial irreducible unitary representation of G associated with λ ∈ P + , λ = 0 and with a K-invariant vector u λ ∈ H λ with u λ 2 = 1. For each n ∈ N, decompose the n-th tensor power (π ⊗,n λ , H ⊗,n λ ) into its irreducible components and consider the orthogonal projections p τn : H ⊗,n λ → H τn . Then for all n ∈ N, Proof. We proceed by induction. In fact, the case n = 1 is trivial. For n → n + 1, we start with (3.2) and the associated orthogonal projections p τn : H ⊗,n λ → H τn . We now decompose the products π τn ⊗ π λ and obtain Notice that here the sum τn µ k,n corresponds to the sum τn+1 of the lemma with n + 1 instead of n. We now consider the orthogonal projections p µ k,n : H ⊗,n+1 λ → H µ k,n . Then where |c| = 1, and thus This fact, the assumption of our induction and the definition of the convolution now readily imply the assertion of the lemma for n + 1.
We shall prove below that under a natural moment condition on a probability measure ν ∈ M 1 (P + ), the C-valued random variables 1 √ n S d,p n converge in distribution for n → ∞. In order to identify the limit distribution µ = µ(d, p, ν) ∈ M 1 (C) in this central limit theorem, we need some further preparations.
3.4 Bessel convolutions on C and Laguerre ensembles. As described in Section 2, the Bessel functions ϕ p λ with λ ∈ C make up the set of bounded spherical functions of the Euclidean symmetric space (F). Thus in the notion of [BH] and [J], the chamber C ∼ = G 0 //K with the associated double coset convolution • d,p is a commutative double coset hypergroup with the functions ϕ p λ as (bounded) hypergroup characters. We now introduce the probability measures on the Weyl chamber C, with the normalization constant The measure ρ d,p ∈ M 1 (C) is well-known in the random matrix theory of so-called Laguerre-or β-ensembles as it is the distribution of the singular values of a M p,q (F)-valued random variable whose reals and imaginary parts of all entries are i.i.d. and standard normally distributed. This fact is well-known; it can also be derived from the Haar measure of the double coset hypergroups (C, • d,p ) in Section 4.1 of [R1]. Moreover, having this group-theoretic interpretation in mind, one easily obtains the following well-known relation from the Fourier transform of a multivariate standard normal distribution: (3.4) see Propos. XV.2.1 of [FK] or [V2]. This identification of the spherical Fourier-transform of ρ d,p will be essential in the following for the central limit theorem.
The probability measure ρ d,p appears in the CLT below as limit up to some scaling parameter σ 2 = σ 2 (ν, p, d), which admits an interpretation as a variance parameter. For the description of σ 2 , we need the so-called moment functions on (P + , * d,p ) up to order two. For the general theory of such moment functions and their applications to limit theorems for random walks on hypergroups we refer to Ch. 7 of [BH], [Z], and references there. The moment functions are characterized by additive functional equations similar to the multiplicative ones for hypergroup characters. They are usually defined in terms of (partial) derivatives with respect to the spectral variables at the identity character.
In our examples, the identity corresponds to x = 0 ∈ C. This motivates the following definition.
3.5 Definition. Let p ∈ [q, ∞[ be fixed. For k, l = 1, . . . , q we define the moment functions m k , m k,l : P + → R of the first and second order by Let us collect some properties of these moment functions.
(2) The functions m k,k are independent of k = 1, . . . , q, and the function m := m 1,1 : P + → R is a quadratic polynomial of the form with suitable coefficients a r,s , b r . In particular, m(0) = 0.
Proof. The Jacobi polynomials R λ (x) are invariant under the Weyl group of type B acting in the variable x. In particular, R λ (x 1 , . . . , x q ) is even in each x i , and this gives part (1). Moreover, as R λ (x 1 , . . . , x q ) is invariant under permutations of the x i , the function m k,k is independent of k. We now study m := m 1,1 more closely. We start with the case p > 2q − 1. In this case we obtain from the integral representation (2.5) that with the power function A short calculation, using that ∆ λ/2 (g 0 (u, w)) = 1, gives Formulas (3.5) and (3.6) now imply that m is a quadratic polynomial as claimed, with linear terms The coefficients a r,s are obtained from the Taylor expansion (2.9) of ∆ r (g ix (u, w)). They are given by a r,s := Bq×U(q,F) tr r (u * (w * P 1 + P 1 · w)u) · tr s (u * (w * P 1 + P 1 · w)u) dm p (w)du (3.7) with the diagonal matrix P 1 := diag(1, 0, . . . , 0) ∈ M q (F). This proves that m is a quadratic polynomial for p > 2q − 1. The case p ≥ q follows by analytic continuation. Finally, for the proof of part (3) we observe that for λ, τ ∈ P + , Notice that the last equality follows from part (1) and R λ (0) = 1.
In fact, this can be easily derived from the definition of m and the explicit formulas for the classical Jacobi polynomials in (2.10) and (2.11). Moreover, it can be also derived from the proof of part (2) of the preceding lemma and a direct elementary computation of a 1,1 and b 1 for q = 1 there.
We shall need the following variant of Lemma 3.6(2) on the growth of m for p ∈ [q, ∞[:
Proof. Let again W denote the Weyl group of type BC q . We introduce the normalized W -invariant orbit sums Then the Jacobi polynomials R p λ can be written as linear combinations of such orbit sums. It follows from the considerations in Section 11 of [M] that for non-negative multiplicity values, the expansion coefficients are all non-negative. That is, Here ≤ denotes the dominance order on P + given by µ Notice that |µ k | ≤ λ k ≤ λ 1 for each µ ∈ W λ. We thus obtain, independently of x ∈ A 0 , Further, if µ ∈ P + with µ ≤ λ, then µ 1 ≤ λ 1 and therefore In the same way, We also need some further properties of the moment function m. We here have to restrict our attention to the case p ∈ {q, q + 1, . . . , 2q − 1}∪]2q − 1, ∞[. We shall assume this restriction from now on. We expect that the results below are also valid for all p ∈ [q, ∞[.
Proof. For the proof of (1), we first consider the case p ∈]2q − 1, ∞[ and conclude from the definition of the a r,s in the proof in Lemma 3.6 that A is symmetric, and that for all τ ∈ R q , are linearly independent for r = 1, . . . q. This shows that τ T Aτ > 0 for all τ ∈ R q with τ = 0 as claimed. The case of integers p ≥ q can be handled in a similar way by using a modified version of integral representation (3.7) for a r,s which is based on Remark 2.3 instead of Proposition 2.1. For the proof of part (2) we proceed as in the proof of Lemma 3.8 and write Thus for λ ∈ P + \ {0}, For the proof of part (3), we use (1) and write m(λ) as with some positive definite matrix A and some b ∈ R q . We thus find constants c, d > 0 such that m(λ) − cλ 2 1 > 0 holds for all λ ∈ P + with λ 1 ≥ d. As there are only finitely many λ ∈ P + with λ 1 < d, we conclude from part (2) that there exists some C 1 > 0 with m(λ) − cλ 2 1 > 0 for all λ ∈ P + with λ = 0.
3.10 Remark. The nonnegativity of m(λ) in Lemma 3.9(2) can be easily established directly. In fact, assume that m(λ) < 0 for some λ ∈ P + . Then the Taylor formula implies that R λ (x) > 1 for some x close to 0, and thus by the Weyl group invariance of R λ , for some x ∈ A 0 . But this contradicts the fact that R λ ∞ ≤ 1 on A 0 , which is a consequence of Proposition 2.1. However, we have no different proof for the strict positivity of m(λ) for λ = 0 than the one given in Lemma 3.9(2) .
We next use the moment function m in order to define a modified variance of measures ν ∈ M 1 (P + ) depending on the underlying convolution * d,p . This modified variance will appear in the CLT below.
Proof. As ν has finite second moments, we conclude form Lemma 3.8 that for all k, l = 1, . . . , q, the series converge uniformly with respect to x ∈ R q . This implies that F ν(x) is twice continuously differentiable on R q , and the partial derivatives commute with the summation. The derivatives at x = 0 are now obtained by Lemma 3.6.
We now turn to random walks (S d,p n ) n≥0 on (P + , * d,p ) associated with some admissible ν ∈ M 1 p (P + ). It is well-known (see e.g. Section 7.3 of [BH]) that the additive functional equation for m in Lemma 3.6(3) leads to relations between the modified variance of ν and random walks associated with ν.
(2) Let (S d,p n ) n≥0 be a random walk on (P + , * d,p ) associated with the measure ν ∈ M 1 p (P + ) with finite second moments. Then, for all integers n ≥ 0, the expectation of m(S d,p n ) satisfies E(m(S d,p n )) = nσ 2 (ν), and the process (m(S d,p n ) − nσ 2 (ν))) n≥0 is a martingale with respect to the canonical filtration of (S d,p n ) n≥0 . Proof. Part (1) follows easily from Lemma 3.6(3); c.f. Section 7.3.7 of [BH]. Moreover, the first assertion of (2) follows from (1) by induction. For the proof of the second statement in (2) we refer to Proposition 7.3.19 of [BH].
Proof. Fix x ∈ C. Let n ∈ N be large enough such that x/ √ n ∈ A 0 . By Section 3.2, S d,p n has the distribution ν (n) . We thus obtain from the multiplicativity of the spherical Fourier transform of probability measures on (P + , * d,p ), Lemma 3.12, the qualitative Taylor formula, and from the properties of the moment functions in Lemma 3.6 that for n → ∞, E(R S d,p n (x/ √ n)) = (F ν(x/ √ n)) n = 1 − σ 2 2n (x 2 1 + . . . x 2 q ) + o(1/n) n (3.8) −→ e −(x 2 1 +...x 2 q )σ 2 /2 . Now fix ǫ > 0. By Lemma 3.14, there is so a > 0 such that for all n ∈ N, P (S d,p n ∈ K √ n·a ) ≤ ǫ. We now conclude from the Mehler-Heine formula 2.4 that for all λ ∈ K √ n·a , that is, λ ∈ P + with λ 1 ≤ a √ n, whenever n is sufficiently large. As |R p λ (x/ √ n)| ≤ 1 and | ϕ p λ (x/ √ n)| ≤ 1, we thus have for n sufficiently large. Together with (3.8) and the identity ϕ p cy (x) = ϕ p y (cx) for c > 0 and x, y ∈ C, this implies that for all x ∈ C, From this limit, equation (3.4) and Levy's continuity theorem for the spherical Fourier transform on the double coset hypergroup (C, • d,p ) (see e.g. Section 4.2 of [BH]), we now infer that S d,p n / √ σ 2 n converges in distribution to ρ d,p as claimed.
Theorem 1.1 in the introduction is an immediate consequence from Theorem 3.15 and Lemma 3.3. We also remark that the methods of the preceding proof lead with some additional technical effort to rates of convergence in the CLT; see [G], [V1] for the rank one case.
We finish this paper with a strong law of large numbers; it follows easily from the preceding properties of the moment function m, in combination with strong laws of large numbers for random walks on commutative hypergroups in Section 7.3 of [BH] and in [Z].
3.16 Theorem. Let ν ∈ M 1 p (P + ) be admissible with with finite second moments, and let (S d,p n ) n≥0 be an associated random walk on (P + , * d,p ). Then for all ǫ > 1/2, S n /n ǫ → 0 almost surely.
Proof. Consider first the hypergroup case with an integer p ≥ q. By Lemmata 3.6 and 3.9, all conditions of Theorem 7.3.26 in [BH] are satisfied for the time-homogeneous random walk (S d,p n ) n≥0 , the sequence (r n := n 2ǫ ) n≥1 , and the moment function m instead of m 2 in [BH]. This theorem now yields that m(S d,p n )/n 2ǫ tends to 0 almost surely, and Lemma 3.9 proves the claim. An inspection of the details in the proof of Theorem 7.3.26 in [BH] shows that this theorem is also available for all p and admissible ν ∈ M 1 p (P + ) which proves the theorem in general.