Operator-Valued and Multivariate Free Berry-Esseen Theorems

Abstract

We address the question of a Berry-Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting operator-valued semicircular element. Since we have to deal with in general non-self-adjoint operators, we introduce the notion of matrix-valued resolvent sets and study the behavior of Cauchy transforms on them.

2010 Mathematics Subject Classification. 46L54, 60F05, 47A10.

1 Introduction

In classical probability theory the famous Berry-Esseen theorem gives a quantitative statement about the order of convergence in the central limit theorem. It states in its simplest version: If $(X_{i})_{i\in \mathbb{N}}$ is a sequence of independent and identically distributed random variables with mean 0 and variance 1, then the distance between $S_{n} := \frac{1} {\sqrt{n}}(X_{1} +\ldots +X_{n})$ and a normal variable γ of mean 0 and variance 1 can be estimated in terms of the Kolmogorov distance $\Delta$ by

$$\Delta (S_{n},\gamma ) \leq C \frac{1} {\sqrt{n}}\rho ,$$

where C is a constant and ρ is the absolute third moment of the variables X _i . The question for a free analogue of the Berry-Esseen estimate in the case of one random variable was answered by Chistyakov and Götze in [2] (and independently, under the more restrictive assumption of compact support of the X _i , by Kargin [10]): If $(X_{i})_{i\in \mathbb{N}}$ is a sequence of free and identically distributed variables with mean 0 and variance 1, then the distance between $S_{n} := \frac{1} {\sqrt{n}}(X_{1} +\ldots +X_{n})$ and a semicircular variable s of mean 0 and variance 1 can be estimated as

$$\Delta (S_{n},s) \leq c\frac{\vert m_{3}\vert + \sqrt{m_{4}}} {\sqrt{n}} ,$$

where c > 0 is an absolute constant and m ₃ and m ₄ are the third and fourth moment, respectively, of the X _i .

In this paper we want to present an approach to a multivariate version of a free Berry-Esseen theorem. The general idea is the following: Since there is up to now no suitable replacement of the Kolmorgorov metric in the multivariate case, we will, in order to describe the speed of convergence of a d-tuple $(S_{n}^{(1)},\ldots ,S_{n}^{(d)})$ of partial sums to the limiting semicircular family $(s_{1},\ldots ,s_{d})$, consider the speed of convergence of $p(S_{n}^{(1)},\ldots ,S_{n}^{(d)})$ to $p(s_{1},\ldots ,s_{d})$ for any self-adjoint polynomial p in d non-commuting variables. By using the linearization trick of Haagerup and Thorbjørnsen [5, 6], we can reformulate this in an operator-valued setting, where we will state an operator-valued free Berry-Esseen theorem. Because estimates for the difference between scalar-valued Cauchy transforms translate by results of Bai [1] to estimates with respect to the Kolmogorov distance, it is convenient to describe the speed of convergence in terms of Cauchy transforms. On the level of deriving equations for the (operator-valued) Cauchy transforms we can follow ideas which are used for dealing with speed of convergence questions for random matrices; here we are inspired in particular by the work of Götze and Tikhomirov [4], but see also [1].

Since the transition from the multivariate to the operator-valued setting leads to operators which are, even if we start from self-adjoint polynomials p, in general not self-adjoint, we have to deal with (operator-valued) Cauchy transforms defined on domains different from the usual ones. Since most of the analytic tools fail in this generality, we have to develop them along the way.

As a first step in this direction, the present paper (which is based on the unpublished preprint [13]) leads finally to the proof of the following theorem:

Theorem 1.1.

Let $(\mathcal{C},\tau )$ be a non-commutative C ^∗ -probability space with τ faithful and put $\mathcal{A} := \mathrm{M}_{m}(\mathbb{C}) \otimes \mathcal{C}$ and E := id ⊗ τ. Let $(X_{i})_{i\in \mathbb{N}}$ be a sequence of non-zero elements in the operator-valued probability space $(\mathcal{A},E)$. We assume:

• All X _i ’s have the same ∗-distribution with respect to E and their first moments vanish, i.e. E[X _i ] = 0.

• The X _i are ∗-free with amalgamation over $\mathrm{M}_{m}(\mathbb{C})$ (which means that the ∗-algebras $\mathcal{X}_{i}$ , generated by $\mathrm{M}_{m}(\mathbb{C})$ and X _i , are free with respect to E).

• We have $\sup _{i\in \mathbb{N}}\|X_{i}\| < \infty$.

Then the sequence $(S_{n})_{n\in \mathbb{N}}$ defined by

$$S_{n} := \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}X_{ i},\qquad n \in \mathbb{N}$$

converges to an operator-valued semicircular element s. Moreover, we can find κ > 0, c > 1, C > 0 and $N \in \mathbb{N}$ such that

$$\|G_{s}(b) - G_{S_{n}}(b)\| \leq C \frac{1} {\sqrt{n}}\|b\|\qquad \mbox{ for all $b \in \Omega $ and $n \geq N$},$$

where

$$\Omega :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\kappa ,\ \|b\| \cdot \| {b}^{-1}\| < c\Big\}$$

and where G_s and $G_{S_{n}}$ denote the operator-valued Cauchy transforms of s and of S_n, respectively.

Applying this operator-valued statement to our multivariate problem gives the following main result on a multivariate free Berry Esseen theorem.

Theorem 1.2.

Let $(x_{i}^{(k)})_{k=1}^{d}$, $i \in \mathbb{N}$ , be free and identically distributed sets of d self-adjoint non-zero random variables in some non-commutative C ^∗ -probability space $(\mathcal{C},\tau )$ , with τ faithful, such that the conditions

$$\tau (x_{i}^{(k)}) = 0\qquad \mbox{ for $k = 1,\ldots ,d$ and all $i \in \mathbb{N}$}$$

and

$$\sup _{i\in \mathbb{N}}\max _{k=1,\ldots ,d}\|x_{i}^{(k)}\| < \infty $$

are fulfilled. We denote by $\Sigma = (\sigma _{k,l})_{k,l=1}^{d}$, where $\sigma _{k,l} :=\tau (x_{i}^{(k)}x_{i}^{(l)})$, their joint covariance matrix. Moreover, we put

$$S_{n}^{(k)} := \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}x_{ i}^{(k)}\qquad \mbox{ for $k = 1,\ldots ,d$ and all $n \in \mathbb{N}$}.$$

Then $(S_{n}^{(1)},\ldots ,S_{n}^{(d)})$ converges in distribution to a semicircular family $(s_{1},\ldots ,s_{d})$ of covariance $\Sigma$. We can quantify the speed of convergence in the following way. Let p be a (not necessarily self-adjoint) polynomial in d non-commutating variables and put

$$P_{n} := p(S_{n}^{(1)},\ldots ,S_{ n}^{(d)})\qquad \mbox{ and}\qquad P := p(s_{ 1},\ldots ,s_{d}).$$

Then, there are constants C > 0, R > 0 and $N \in \mathbb{N}$ (depending on the polynomial) such that

$$\vert G_{P}(z) - G_{P_{n}}(z)\vert \leq C \frac{1} {\sqrt{n}}\qquad \mbox{ for all $\vert z\vert > R$ and $n \geq N$},$$

where G_P and $G_{P_{n}}$ denote the scalar-valued Cauchy transform of P and of P_n, respectively.

In the case of a self-adjoint polynomial p, we can consider the distribution measures μ _n and μ of the operators P _n and P from above, which are probability measures on $\mathbb{R}$. Moreover, let $\mathcal{F}_{\mu _{n}}$ and $\mathcal{F}_{\mu }$ be their cumulative distribution functions. In order to deduce estimates for the Kolmogorov distance

$$\Delta (\mu _{n},\mu ) =\sup _{x\in \mathbb{R}}\vert \mathcal{F}_{\mu _{n}}(x) -\mathcal{F}_{\mu }(x)\vert $$

one has to transfer the estimate for the difference of the scalar-valued Cauchy transforms of P _n and P from near infinity to a neighborhood of the real axis. A partial solution to this problem was given in the appendix of [14], which we will recall in Sect. 4. But this leads to the still unsolved question, whether $p(s_{1},\ldots ,s_{d})$ has a continuous density. We conjecture that the latter is true for any self-adjoint polynomial in free semicirculars, but at present we are not aware of a proof of that statement.

The paper is organized as follows. In Sect. 2 we recall some basic facts about holomorphic functions on domains in Banach spaces. The tools to deal with matrix-valued Cauchy transform will be presented in Sect. 3. Section 4 is devoted to the proof of Theorems 1.1 and 1.2.

2 Holomorphic Functions on Domains in Banach Spaces

For reader’s convenience, we briefly recall the definition of holomorphic functions on domains in Banach spaces and we state the theorem of Earle-Hamilton, which will play a major role in the subsequent sections.

Definition 2.1.

Let $(X,\|\cdot \|_{X})$, $(Y,\|\cdot \|_{Y })$ be two complex Banach spaces and let D ⊆ X be an open subset of X. A function f : D → Y is called

• Strongly holomorphic, if for each x ∈ D there exists a bounded linear mapping Df(x) : X → Y such that

  $$\lim _{y\rightarrow 0}\frac{\|f(x + y) - f(x) - Df(x)y\|_{Y }} {\|y\|_{X}} = 0.$$

• Weakly holomorphic, if it is locally bounded and the mapping

  $$\lambda \mapsto \phi (f(x +\lambda y))$$

  is holomorphic at λ = 0 for each x ∈ D, y ∈ Y and all continuous linear functionals $\phi : Y \rightarrow \mathbb{C}$.

An important theorem due to Dunford says, that a function on a domain (i.e. an open and connected subset) in a Banach space is strongly holomorphic if and only if it is weakly holomorphic. Hence, we do not have to distinguish between both definitions.

Definition 2.2.

Let D be a nonempty domain in a complex Banach space $(X,\|\cdot \|)$ and let f : D → D be a holomorphic function. We say, that f(D) lies strictly inside D, if there is some ε > 0 such that

$$B_{\epsilon }(f(x)) \subseteq D\qquad \mbox{ for all $x \in D$}$$

holds, whereby we denote by B _r (y) the open ball with radius r around y.

The remarkable fact, that strict holomorphic mappings are strict contractions in the so-called Carathéodory-Riffen-Finsler metric, leads to the following theorem of Earle-Hamilton (cf. [3]), which can be seen as a holomorphic version of Banach’s contraction mapping theorem. For a proof of this theorem and variations of the statement we refer to [7].

Theorem 2.3 (Earle-Hamilton, 1970). 

Let ∅≠D ⊆ X be a domain in a Banach space $(X,\|\cdot \|)$ and let f : D → D be a bounded holomorphic function. If f(D) lies strictly inside D, then f has a unique fixed point in D.

3 Matrix-Valued Spectra and Cauchy Transforms

The statement of the following lemma is well-known and quite simple. But since it turns out to be extremely helpful, it is convenient to recall it here.

Lemma 3.1.

Let $(A,\|\cdot \|)$ be a complex Banach-algebra with unit 1. If x ∈ A is invertible and y ∈ A satisfies $\|x - y\| <\sigma \frac{1} {\|{x}^{-1}\|}$ for some 0 < σ < 1, then y is invertible as well and we have

$$\|{y}^{-1}\| \leq \frac{1} {1-\sigma }\|{x}^{-1}\|.$$

Proof.

We can easily check that

$$\displaystyle\sum _{n=0}^{\infty }\big{({x}^{-1}(x - y)\big)}^{n}{x}^{-1}$$

is absolutely convergent in A and gives the inverse element of y. Moreover we get

$$\|{y}^{-1}\| \leq \displaystyle\sum _{ n=0}^{\infty }\big{(\|{x}^{-1}\|\|x - y\|\big)}^{n}\|{x}^{-1}\| < \frac{1} {1-\sigma }\|{x}^{-1}\|,$$

which proves the stated estimate. □ 

Let $(\mathcal{C},\tau )$ be a non-commutative C ^∗-probability space, i.e., $\mathcal{C}$ is a unital C ^∗-algebra and τ is a unital state (positive linear functional) on $\mathcal{C}$; we will always assume that τ is faithful. For fixed $m \in \mathbb{N}$ we define the operator-valued C ^∗-probability space $\mathcal{A} := \mathrm{M}_{m}(\mathbb{C}) \otimes \mathcal{C}$ with conditional expectation

$$E := \mathrm{id}_{m}\otimes \tau :\ \mathcal{A}\rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b \otimes c\mapsto \tau (c)b,$$

where we denote by $\mathrm{M}_{m}(\mathbb{C})$ the C ^∗-algebra of all m ×m matrices over the complex numbers $\mathbb{C}$. Under the canonical identification of $\mathrm{M}_{m}(\mathbb{C}) \otimes \mathcal{C}$ with $\mathrm{M}_{m}(\mathcal{C})$ (matrices with entries in $\mathcal{C}$), the expectation E corresponds to applying the state τ entrywise in a matrix. We will also identify $b \in \mathrm{M}_{m}(\mathbb{C})$ with $b \otimes 1 \in \mathcal{A}$.

Definition 3.2.

For $a \in \mathcal{A} = \mathrm{M}_{m}(\mathcal{C})$ we define the matrix-valued resolvent set

$$\rho _{m}(a) :=\{ b \in \mathrm{M}_{m}(\mathbb{C})\mid \mbox{ $b - a$ is invertible in $\mathcal{A}$}\}$$

and the matrix-valued spectrum

$$\sigma _{m}(a) := \mathrm{M}_{m}(\mathbb{C})\setminus \rho _{m}(a).$$

Since the set $\mathrm{GL}(\mathcal{A})$ of all invertible elements in $\mathcal{A}$ is an open subset of $\mathcal{A}$ (cf. Lemma 3.1), the continuity of the mapping

$$f_{a} :\ \mathrm{M}_{m}(\mathbb{C}) \rightarrow \mathcal{A},\ b\mapsto b - a$$

implies, that the matrix-valued resolvent set $\rho _{m}(a) = f_{a}^{-1}(\mathrm{GL}(\mathcal{A}))$ of an element $a \in \mathcal{A}$ is an open subset of $\mathrm{M}_{m}(\mathbb{C})$. Hence, the matrix-valued spectrum σ _m (a) is always closed.

Although the behavior of this matrix-valued generalizations of the classical resolvent set and spectrum seems to be quite similar to the classical case (which is of course included in our definition for m = 1), the matrix valued spectrum is in general not bounded and hence not a compact subset of $\mathrm{M}_{m}(\mathbb{C})$. For example, we have for all $\lambda \in \mathbb{C}$, that

$$\sigma _{m}(\lambda 1) =\{ b \in \mathrm{M}_{m}(\mathbb{C})\mid \lambda \in \sigma _{\mathrm{M}_{m}(\mathbb{C})}(b)\},$$

i.e. σ _m (λ1) consists of all matrices $b \in \mathrm{M}_{m}(\mathbb{C})$ for which λ belongs to the spectrum $\sigma _{\mathrm{M}_{m}(\mathbb{C})}(b)$. Particularly, σ _m (λ1) is unbounded for m ≥ 2.

In the following, we denote by $\mathrm{GL}_{m}(\mathbb{C}) := \mathrm{GL}(\mathrm{M}_{m}(\mathbb{C}))$ the set of all invertible matrices in $\mathrm{M}_{m}(\mathbb{C})$.

Lemma 3.3.

Let $a \in \mathcal{A}$ be given. Then for all $b \in \mathrm{GL}_{m}(\mathbb{C})$ the following inclusion holds:

$$\big\{\lambda b\mid \lambda \in \rho _{\mathcal{A}}({b}^{-1}a)\big\} \subseteq \rho _{ m}(a)$$

Proof.

Let $\lambda \in \rho _{\mathcal{A}}({b}^{-1}a)$ be given. By definition of the usual resolvent set this means that $\lambda 1 - {b}^{-1}a$ is invertible in $\mathcal{A}$. It follows, that

$$\lambda b - a = b\big(\lambda 1 - {b}^{-1}a\big)$$

is invertible as well, and we get, as desired, λb ∈ ρ _m (a). □ 

Lemma 3.4.

For all $0\neq a \in \mathcal{A}$ we have

$$\Big\{b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| < \frac{1} {\|a\|}\Big\} \subseteq \rho _{m}(a)$$

and

$$\sigma _{m}(a) \cap \mathrm{GL}_{m}(\mathbb{C}) \subseteq \Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| \geq \frac{1} {\|a\|}\Big\}.$$

Proof.

Obviously, the second inclusion is a direct consequence of the first. Hence, it suffices to show the first statement.

Let $b \in \mathrm{GL}_{m}(\mathbb{C})$ with $\|{b}^{-1}\| < \frac{1} {\|a\|}$ be given. It follows, that $h := 1 - {b}^{-1}a$ is invertible, because

$$\|1 - h\| =\| {b}^{-1}a\| \leq \| {b}^{-1}\| \cdot \| a\| < 1.$$

Therefore, we can deduce, that also

$$b - a = b\big(1 - {b}^{-1}a\big)$$

(1)

is invertible, i.e. b ∈ ρ _m (a). This proves the assertion. □ 

The main reason to consider matrix-valued resolvent sets is, that they are the natural domains for matrix-valued Cauchy transforms, which we will define now.

Definition 3.5.

For $a \in \mathcal{A}$ we call

$$G_{a} :\ \rho _{m}(a) \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b\mapsto E\big[{(b - a)}^{-1}\big]$$

the matrix-valued Cauchy transform of a.

Note that G _a is a continuous function (and hence locally bounded) and induces for all b ₀ ∈ ρ _m (a), $b \in \mathrm{M}_{m}(\mathbb{C})$ and bounded linear functionals $\phi : \mathcal{A}\rightarrow \mathbb{C}$ a function

$$\lambda \mapsto \phi \big(G_{a}(b_{0} +\lambda b)\big),$$

which is holomorphic in a neighborhood of λ = 0. Hence, G _a is weakly holomorphic and therefore (as we have seen in the previous section) strongly holomorphic as well.

Because the structure of ρ _m (a) and therefore the behavior of G _a might in general be quite complicated, we restrict our attention to a suitable restriction of G _a . In this way, we will get some additional properties of G _a .

The first restriction enables us to control the norm of the matrix-valued Cauchy transform on a sufficiently nice subset of the matrix-valued resolvent set.

Lemma 3.6.

Let $0\neq a \in \mathcal{A}$ be given. For 0 < θ < 1 the matrix valued Cauchy transform G_a induces a mapping

$$G_{a} :\ \Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\theta \cdot \frac{1} {\|a\|}\Big\} \rightarrow \Big\{ b \in \mathrm{M}_{m}(\mathbb{C})\mid \|b\| < \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|}\Big\}.$$

Proof.

Lemma 3.4 (c) tells us, that the open set

$$U :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\theta \cdot \frac{1} {\|a\|}\Big\}$$

is contained in ρ _m (a), i.e. G _a is well-defined on U. Moreover, we get from (1)

$${(b - a)}^{-1} =\big {(1 - {b}^{-1}a\big)}^{-1}{b}^{-1} =\displaystyle\sum _{ n=0}^{\infty }\big{({b}^{-1}a\big)}^{n}{b}^{-1}$$

and hence

$$\|G_{a}(b)\| \leq \| {(b - a)}^{-1}\| \leq \| {b}^{-1}\|\displaystyle\sum _{ n=0}^{\infty }\big{(\|{b}^{-1}\|\|a\|\big)}^{n} < \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|}$$

(2)

for all b ∈ U. This proves the claim. □ 

To ensure, that the range of G _a is contained in $\mathrm{GL}_{m}(\mathbb{C})$, we have to shrink the domain again.

Lemma 3.7.

Let $0\neq a \in \mathcal{A}$ be given. For 0 < θ < 1 and c > 1 we define

$$\Omega :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\theta \cdot \frac{1} {\|a\|},\ \|b\| \cdot \| {b}^{-1}\| < c\Big\}$$

and

$$\Omega ^\prime :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|b\| < \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|}\Big\}.$$

If the condition

$$\frac{\theta } {1-\theta } < \frac{\sigma } {c}$$

is satisfied for some 0 < σ < 1, then the matrix-valued Cauchy transform G_a induces a mapping $G_{a} : \Omega \rightarrow \Omega ^\prime$ and we have the estimates

$$\|G_{a}(b)\| \leq \| {(b - a)}^{-1}\| < \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|}\qquad \mbox{ for all $b \in \Omega $}$$

(3)

and

$$\|G_{a}{(b)}^{-1}\| < \frac{1} {1-\sigma }\cdot \| b\|\qquad \mbox{ for all $b \in \Omega $}.$$

(4)

Proof.

For all $b \in \Omega$ we have

$$G_{a}(b) - {b}^{-1} = E\big[{(b - a)}^{-1} - {b}^{-1}\big] = E\Big[\displaystyle\sum _{ n=1}^{\infty }\big{({b}^{-1}a\big)}^{n}{b}^{-1}\Big],$$

which enables us to deduce

$$\|G_{a}(b) - {b}^{-1}\| \leq \| {b}^{-1}\|\displaystyle\sum _{ n=1}^{\infty }\big{(\|{b}^{-1}\|\|a\|\big)}^{n} \leq \frac{\theta } {1-\theta }\cdot \| {b}^{-1}\| < \frac{\theta } {1-\theta }\cdot \frac{c} {\|b\|} <\sigma \cdot \frac{1} {\|b\|}.$$

Using Lemma 3.1, this implies $G_{a}(b) \in \mathrm{GL}_{m}(\mathbb{C})$ and (4). Since we already know from (2) in Lemma 3.6, that (3) holds, it follows $G_{a}(b) \in \Omega ^\prime$ and the proof is complete. □ 

Remark 3.8.

Since domains of our holomorphic functions should be connected it is necessary to note, that for κ > 0 and c > 1

$$\Omega =\big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\kappa ,\ \|b\| \cdot \| {b}^{-1}\| < c\big\}$$

and for r > 0

$$\Omega ^\prime =\big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|b\| < r\big\}$$

are pathwise connected subsets of $\mathrm{M}_{m}(\mathbb{C})$. Indeed, if $b_{1},b_{2} \in \mathrm{GL}_{m}(\mathbb{C})$ are given, we consider their polar decomposition $b_{1} = U_{1}P_{1}$ and $b_{2} = U_{2}P_{2}$ with unitary matrices $U_{1},U_{2} \in \mathrm{GL}_{m}(\mathbb{C})$ and positive-definite Hermitian matrices $P_{1},P_{2} \in \mathrm{GL}_{m}(\mathbb{C})$ and define (using functional calculus for normal elements in the C ^∗-algebra $\mathrm{M}_{m}(\mathbb{C})$)

$$\gamma :\ [0,1] \rightarrow \mathrm{GL}_{m}(\mathbb{C}),\ t\mapsto U_{1}^{1-t}P_{ 1}^{1-t}U_{ 2}^{t}P_{ 2}^{t}.$$

Then γ fulfills γ(0) = b ₁ and γ(1) = b ₂, and γ([0, 1]) is contained in $\Omega$ and $\Omega ^\prime$ if b ₁, b ₂ are elements of $\Omega$ and $\Omega ^\prime$, respectively.

Since the matrix-valued Cauchy transform is a solution of a special equation (cf. [8, 12]), we will be interested in the following situation:

Corollary 3.9.

Let $\eta : \mathrm{GL}_{m}(\mathbb{C}) \rightarrow \mathrm{M}_{m}(\mathbb{C})$ be a holomorphic function satisfying

$$\|\eta (w)\| \leq M\|w\|\qquad \mbox{ for all $w \in \mathrm{GL}_{m}(\mathbb{C})$}$$

for some M > 0. Moreover, we assume that

$$bG_{a}(b) = 1 +\eta (G_{a}(b))G_{a}(b)\qquad \mbox{ for all $b \in \Omega $}$$

holds. Let 0 < θ,σ < 1 and c > 1 be given with

$$\frac{\theta } {1-\theta } <\sigma \min \Big\{ \frac{1} {c},\ \frac{\|{a\|}^{2}} {M}\Big\}$$

and let $\Omega$ and $\Omega ^\prime$ be as in Lemma 3.7.

Then, for fixed $b \in \Omega$ , the equation

$$bw = 1 +\eta (w)w,\qquad w \in \Omega ^\prime $$

(5)

has a unique solution, which is given by w = G _a (b).

Proof.

Let $b \in \Omega$ be given. For all $w \in \Omega ^\prime$ we get

$$\|\eta (w)\| \leq M\|w\| \leq \frac{\theta } {1-\theta }\cdot \frac{M} {\|a\|}$$

and therefore

$$\|{b}^{-1}\eta (w)\| \leq \| {b}^{-1}\|\|\eta (w)\| \leq \frac{\theta } {1-\theta }\cdot \frac{M} {\|{a\|}^{2}} \cdot \theta <\theta \sigma < 1.$$

This means, that $1 - {b}^{-1}\eta (w)$ and hence b − η(w) is invertible with

$$\displaystyle\begin{array}{rcl} \|{(b -\eta (w))}^{-1}\|& \leq & \|{b}^{-1}\|\|{(1 - {b}^{-1}\eta (w))}^{-1}\| \\ & \leq & \|{b}^{-1}\|\displaystyle\sum _{ n=0}^{\infty }\|{b}^{-1}\eta {(w)\|}^{n} \\ & <& \frac{\theta } {1-\theta \sigma } \cdot \frac{1} {\|a\|}, \\ \end{array}$$

and shows, that we have a well-defined and holomorphic mapping

$$\mathcal{F} :\ \Omega ^\prime \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ w\mapsto {(b -\eta (w))}^{-1}$$

with

$$\|\mathcal{F}(w)\| =\| {(b -\eta (w))}^{-1}\| < \frac{\theta } {1-\theta \sigma }\cdot \frac{1} {\|a\|} < \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|}$$

and therefore $\mathcal{F}(w) \in \Omega ^\prime$.

Now, we want to show that $\mathcal{F}(\Omega ^\prime )$ lies strictly inside $\Omega ^\prime$. We put

$$\epsilon :=\min \Big\{ \frac{1} {2} \cdot \frac{1} {\|b\| +\sigma \| a\|},\ \Big(1 -\frac{1-\theta } {1-\theta \sigma }\Big) \cdot \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|}\Big\} > 0$$

and consider $w \in \Omega ^\prime$ and $u \in \mathrm{M}_{m}(\mathbb{C})$ with $\|u -\mathcal{F}(w)\| <\epsilon$. At first, we get

$$\|b -\eta (w)\| \leq \| b\| +\|\eta (w)\| \leq \| b\| + \frac{M} {\|a\|} \cdot \frac{\theta } {1-\theta } \leq \| b\| +\sigma \| a\|$$

and thus

$$\|u - {(b -\eta (w))}^{-1}\| =\| u -\mathcal{F}(w)\| <\epsilon \leq \frac{1} {2} \cdot \frac{1} {\|b\| +\sigma \| a\|} \leq \frac{1} {2} \cdot \frac{1} {\|b -\eta (w)\|},$$

which shows $u \in \mathrm{GL}_{m}(\mathbb{C})$, and secondly

$$\displaystyle\begin{array}{rcl} \|u\|& =& \|u - {(b -\eta (w))}^{-1}\| +\| \mathcal{F}(w)\| \\ & <& \epsilon +\frac{1-\theta } {1-\theta \sigma }\cdot \frac{\theta } {1-\theta }\cdot \frac{1} {\|a\|} \\ & <& \frac{\theta } {1-\theta } \cdot \frac{1} {\|a\|} \\ \end{array}$$

which shows $u \in \Omega ^\prime$.

Let now $w \in \Omega ^\prime$ be a solution of (5). This implies that

$${w}^{-1}\mathcal{F}(w) = {w}^{-1}{(b -\eta (w))}^{-1} =\big {(bw -\eta (w)w\big)}^{-1} = 1,$$

and hence $\mathcal{F}(w) = w$. Since $\mathcal{F} : \Omega ^\prime \rightarrow \Omega ^\prime$ is holomorphic on the domain $\Omega ^\prime$ and $\mathcal{F}(\Omega ^\prime )$ lies strictly inside $\Omega ^\prime$, it follows by the Theorem of Earle-Hamilton, Theorem 2.3, that $\mathcal{F}$ has exactly one fixed point. Because G _a (b) (which is an element of $\Omega ^\prime$ by Lemma 3.7) solves (5) by assumption and hence is already a fixed point of $\mathcal{F}$, it follows w = G _a (b) and we are done. □ 

Remark 3.10.

Let $(\mathcal{A}^\prime ,E^\prime )$ be an arbitrary operator-valued C ^∗-probability space with conditional expectation $E^\prime : \mathcal{A}^\prime \rightarrow \mathrm{M}_{m}(\mathbb{C})$. This provides us with a unital (and continuous) ∗ -embedding $\iota : \mathrm{M}_{m}(\mathbb{C}) \rightarrow \mathcal{A}^\prime$. In this section, we only considered the special embedding

$$\iota :\ \mathrm{M}_{m}(\mathbb{C}) \rightarrow \mathcal{A},\ b\mapsto b \otimes 1,$$

which is given by the special structure $\mathcal{A} = \mathrm{M}_{m}(\mathbb{C}) \otimes \mathcal{C}$. But we can define matrix-valued resolvent sets, spectra and Cauchy transforms also in this more general framework. To be more precise, we put for all $a \in \mathcal{A}^\prime$

$$\rho _{m}(a) :=\{ b \in \mathrm{M}_{m}(\mathbb{C})\mid \mbox{ $\iota (b) - a$ is invertible in $\mathcal{A}^\prime $}\}$$

and $\sigma _{m}(a) := \mathrm{M}_{m}(\mathbb{C})\setminus \rho _{m}(a)$ and

$$G_{a} :\ \rho _{m}(a) \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b\mapsto E^\prime \big[{(\iota (b) - a)}^{-1}\big].$$

We note, that all the results of this section stay valid in this general situation.

4 Multivariate Free Central Limit Theorem

4.1 Setting and First Observations

Let $(X_{i})_{i\in \mathbb{N}}$ be a sequence in the operator-valued probability space $(\mathcal{A},E)$ with $\mathcal{A} = \mathrm{M}_{m}(\mathcal{C}) = \mathrm{M}_{m}(\mathbb{C}) \otimes \mathcal{C}$ and E = id ⊗ τ, as defined in the previous section. We assume:

• All X _i ’s have the same ∗ -distribution with respect to E and their first moments vanish, i.e. E[X _i ] = 0.

• The X _i are ∗ -free with amalgamation over $\mathrm{M}_{m}(\mathbb{C})$ (which means that the ∗ -algebras $\mathcal{X}_{i}$, generated by $\mathrm{M}_{m}(\mathbb{C})$ and X _i , are free with respect to E).

• We have $\sup _{i\in \mathbb{N}}\|X_{i}\| < \infty$.

If we define the linear (and hence holomorphic) mapping

$$\eta :\ \mathrm{M}_{m}(\mathbb{C}) \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b\mapsto E[X_{i}bX_{i}],$$

we easily get from the continuity of E, that

$$\|\eta (b)\| \leq \Big {(\sup _{i\in \mathbb{N}}\|X_{i}\|\Big)}^{2}\|b\|\qquad \mbox{ for all $b \in \mathrm{M}_{ m}(\mathbb{C})$}$$

holds. Hence we can find M > 0 such that $\|\eta (b)\| < M\|b\|$ holds for all $b \in \mathrm{M}_{m}(\mathbb{C})$. Moreover, we have for all $k \in \mathbb{N}$ and all $b_{1},\ldots ,b_{k} \in \mathrm{M}_{m}(\mathbb{C})$

$$\sup _{i\in \mathbb{N}}\|E[X_{i}b_{1}X_{i}\ldots b_{k}X_{i}]\| \leq \Big {(\sup _{i\in \mathbb{N}}\|X_{i}\|\Big)}^{k+1}\|b_{ 1}\|\cdots \|b_{k}\|.$$

Since $(X_{i})_{i\in \mathbb{N}}$ is a sequence of centered free non-commutative random variables, Theorem 8.4 in [15] tells us that the sequence $(S_{n})_{n\in \mathbb{N}}$ defined by

$$S_{n} := \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}X_{ i},\qquad n \in \mathbb{N}$$

converges to an operator-valued semicircular element s. Moreover, we know from Theorem 4.2.4 in [12] that the operator-valued Cauchy transform G _s satisfies

$$bG_{s}(b) = 1 +\eta (G_{s}(b))G_{s}(b)\qquad \mbox{ for all $b \in U_{r}$},$$

where we put $U_{r} :=\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| < r\} \subseteq \rho _{m}(s)$ for all suitably small r > 0.

By Proposition 7.1 in [9], the boundedness of the sequence $(X_{i})_{i\in \mathbb{N}}$ guarantees boundedness of $(S_{n})_{n\in \mathbb{N}}$ as well. In order to get estimates for the difference between the Cauchy transforms G _s and $G_{S_{n}}$ we will also need the fact, that $(S_{n})_{n\in \mathbb{N}}$ is bounded away from 0. The precise statement is part of the following lemma, which also includes a similar statement for

$$S_{n}^{[i]} := S_{ n} - \frac{1} {\sqrt{n}}X_{i} = \frac{1} {\sqrt{n}}\displaystyle\sum _{{ j=1 \atop j\not =i} }^{n}X_{ j}\qquad \mbox{ for all $n \in \mathbb{N}$ and $1 \leq i \leq n$}.$$

Lemma 4.1.

In the situation described above, we have for all $n \in \mathbb{N}$ and all 1 ≤ i ≤ n

$$\|S_{n}\| {\geq \|\alpha \|}^{\frac{1} {2} }\qquad \mbox{ and}\qquad \|S_{n}^{[i]}\| \geq {\sqrt{1 - \frac{1} {n}}\|\alpha \|}^{\frac{1} {2} },$$

where $\alpha := E[X_{i}^{{\ast}}X_{i}] \in \mathrm{M}_{m}(\mathbb{C})$ .

Proof.

By the ∗ -freeness of $X_{1},X_{2},\ldots$, we have

$$E[X_{i}^{{\ast}}X_{ j}] = E[X_{i}^{{\ast}}] \cdot E[X_{ j}] = 0,\qquad \mbox{ for $i\not =j$}$$

and thus

$$\|S{_{n}\|}^{2} =\| S_{ n}^{{\ast}}S_{ n}\| \geq \| E[S_{n}^{{\ast}}S_{ n}]\| = \frac{1} {n}\bigg\|\displaystyle\sum _{i,j=1}^{n}E[X_{ i}^{{\ast}}X_{ j}]\bigg\| =\|\alpha \|.$$

Similarly

$$\displaystyle\begin{array}{rcl} \|S_{n}^{[i]}\|^{2}& =& \|{(S_{ n}^{[i]})}^{{\ast}}S_{ n}^{[i]}\| \\ & \geq & \|E[{(S_{n}^{[i]})}^{{\ast}}S_{ n}^{[i]}]\| \\ & =& \bigg\|E[S_{n}^{{\ast}}S_{ n}] - \frac{1} {n}E[X_{i}^{{\ast}}X_{ i}]\bigg\| \\ & =& \frac{n - 1} {n} \|\alpha \|, \\ \end{array}$$

which proves the statement. □ 

We define for $n \in \mathbb{N}$

$$R_{n} :\ \rho _{m}(S_{n}) \rightarrow \mathcal{A},\ b\mapsto \big{(b - S_{n}\big)}^{-1}$$

and for $n \in \mathbb{N}$ and 1 ≤ i ≤ n

$$R_{n}^{[i]} :\ \rho _{ m}(S_{n}^{[i]}) \rightarrow \mathcal{A},\ b\mapsto \big{(b - S_{ n}^{[i]}\big)}^{-1}.$$

Lemma 4.2.

For all $n \in \mathbb{N}$ and 1 ≤ i ≤ n we have

$$R_{n}(b) = R_{n}^{[i]}(b) + \frac{1} {\sqrt{n}}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b) + \frac{1} {n}R_{n}(b)X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)$$

(6)

and

$$R_{n}(b) = R_{n}^{[i]}(b) + \frac{1} {\sqrt{n}}R_{n}^{[i]}(b)X_{ i}R_{n}(b)$$

(7)

for all $b \in \rho _{m}(S_{n}) \cap \rho _{m}(S_{n}^{[i]})$ .

Proof.

We have

$$\displaystyle\begin{array}{rcl} \big(b - S_{n}\big)R_{n}(b)\big(b - S_{n}^{[i]}\big)& =& \ b - S_{ n}^{[i]} \\ & =& \ \big(b - S_{n}\big) + \frac{1} {\sqrt{n}}\big(b - S_{n}^{[i]}\big)R_{ n}^{[i]}(b)X_{ i} \\ & =& \ \big(b - S_{n}\big) + \frac{1} {\sqrt{n}}\big(b - S_{n}\big)R_{n}^{[i]}(b)X_{ i} + \frac{1} {n}X_{i}R_{n}^{[i]}(b)X_{ i},\\ \end{array}$$

which leads, by multiplication with $R_{n}(b) = {(b - S_{n})}^{-1}$ from the left and with $R_{n}^{[i]}(b) = {(b - S_{n}^{[i]})}^{-1}$ from the right, to (6).

Moreover, we have

$$\big(b - S_{n}^{[i]}\big)R_{ n}(b)\big(b - S_{n}\big) = b - S_{n}^{[i]} =\big (b - S_{ n}\big) + \frac{1} {\sqrt{n}}X_{i},$$

which leads, by multiplication with $R_{n}(b) = {(b - S_{n})}^{-1}$ from the right and with $R_{n}^{[i]}(b) = {(b - S_{n}^{[i]})}^{-1}$ from the left, to equation (7). □ 

Obviously, we have

$$G_{n} := G_{S_{n}} = E \circ R_{n}\qquad \mbox{ and}\qquad G_{n}^{[i]} := G_{ S_{n}^{[i]}} = E \circ R_{n}^{[i]}.$$

4.2 Proof of the Main Theorem

During this subsection, let 0 < θ, σ < 1 and c > 1 be given, such that

$$\frac{\theta } {1-\theta } <\sigma \min \Big\{ \frac{1} {c},\ \frac{\|\alpha \|} {M}\Big\}$$

(8)

holds. For all $n \in \mathbb{N}$ we define

$$\kappa _{n} :=\theta \min \Big\{ \frac{1} {\|s\|}, \frac{1} {\|S_{n}\|}, \frac{1} {\|S_{n}^{[1]}\|},\ldots , \frac{1} {\|S_{n}^{[n]}\|}\Big\}$$

and

$$\Omega _{n} :=\big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\kappa _{ n},\ \|b\| \cdot \| {b}^{-1}\| < c\big\}.$$

Lemma 3.4 shows, that $\Omega _{n}$ is a subset of ρ _m (S _n ).

Theorem 4.3.

For all $2 \leq n \in \mathbb{N}$ the function G _n satisfies the following equation

$$\Lambda _{n}(b)G_{n}(b) = 1 +\eta (G_{n}(b))G_{n}(b),\qquad b \in \Omega _{n},$$

where

$$\Lambda _{n} :\ \Omega _{n} \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b\mapsto b - \Theta _{n}(b)G_{n}{(b)}^{-1},$$

with a holomorphic function

$$\Theta _{n} :\ \Omega _{n} \rightarrow \mathrm{M}_{m}(\mathbb{C})$$

satisfying

$$\sup _{b\in \Omega _{n}}\|\Theta _{n}(b)\| \leq \frac{C} {\sqrt{n}}$$

with a constant C > 0, independent of n.

Proof.

• (i) Let $n \in \mathbb{N}$ and b ∈ ρ _m (S _n ) be given. Then we have

  $$S_{n}R_{n}(b) = bR_{n}(b) - (b - S_{n})R_{n}(b) = bR_{n}(b) - 1$$

  and hence

  $$E[S_{n}R_{n}(b)] = E\big[bR_{n}(b) - 1\big] = bG_{n}(b) - 1.$$

• (ii) Let $n \in \mathbb{N}$ be given. For all

  $$b \in \rho _{m,n} :=\rho _{m}(S_{n}) \cap \displaystyle\bigcap _{i=1}^{n}\rho _{ m}(S_{n}^{[i]})$$

  we deduce from the formula in (6), that

  $$\displaystyle\begin{array}{rcl} & & E[S_{n}R_{n}(b)] = \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}E[X_{ i}R_{n}(b)] \\ & & \quad = \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}\bigg(E\big[X_{ i}R_{n}^{[i]}(b)\big] + \frac{1} {\sqrt{n}}E\big[X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big] \\ & & \qquad + \frac{1} {n}E\big[X_{i}R_{n}(b)X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big]\bigg) \\ & & \quad = \frac{1} {n}\displaystyle\sum _{i=1}^{n}\bigg(E\big[X_{ i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big] + \frac{1} {\sqrt{n}}E\big[X_{i}R_{n}(b)X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big]\bigg) \\ & & \quad = \frac{1} {n}\displaystyle\sum _{i=1}^{n}\bigg(E\big[X_{ i}G_{n}^{[i]}(b)X_{ i}\big]G_{n}^{[i]}(b) + \frac{1} {\sqrt{n}}E\big[X_{i}R_{n}(b)X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big]\bigg) \\ & & \quad = \frac{1} {n}\displaystyle\sum _{i=1}^{n}\Big(\eta (G_{ n}^{[i]}(b))G_{ n}^{[i]}(b) + r_{ n,1}^{[i]}(b)\Big), \\ \end{array}$$

  where

  $$r_{n,1}^{[i]} :\rho _{ m}(S_{n}) \cap \rho _{m}(S_{n}^{[i]})\,\rightarrow \,\mathrm{M}_{ m}(\mathbb{C}),\ b\,\mapsto \, \frac{1} {\sqrt{n}}E\big[X_{i}R_{n}(b)X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big].$$

  There we used the fact, that, since the $(X_{j})_{j\in \mathbb{N}}$ are free with respect to E, also X _i is free from R _n ^[i], and thus we have

  $$E\big[X_{i}R_{n}^{[i]}(b)\big] = E[X_{ i}]E\big[R_{n}^{[i]}(b)\big] = 0$$

  and

  $$E\big[X_{i}R_{n}^{[i]}(b)X_{ i}R_{n}^{[i]}(b)\big] = E\big[X_{ i}E\big[R_{n}^{[i]}(b)\big]X_{ i}\big]E\big[R_{n}^{[i]}(b)\big].$$

• (iii) Taking (7) into account, we get for all $n \in \mathbb{N}$ and 1 ≤ i ≤ n

  $$G_{n}(b) = E\big[R_{n}(b)\big] = E\big[R_{n}^{[i]}(b)\big] + \frac{1} {\sqrt{n}}E\big[R_{n}^{[i]}(b)X_{ i}R_{n}(b)\big] = G_{n}^{[i]}(b) - r_{ n,2}^{[i]}(b)$$

  and therefore

  $$G_{n}^{[i]}(b) = G_{ n}(b) + r_{n,2}^{[i]}(b)$$

  for all $b \in \rho _{m}(S_{n}) \cap \rho _{m}(S_{n}^{[i]})$, where we put

  $$r_{n,2}^{[i]} :\ \rho _{ m}(S_{n}) \cap \rho _{m}(S_{n}^{[i]}) \rightarrow \mathrm{M}_{ m}(\mathbb{C}),\ b\mapsto - \frac{1} {\sqrt{n}}E\big[R_{n}^{[i]}(b)X_{ i}R_{n}(b)\big].$$

• (iv) The formula in (iii) enables us to replace G _n ^[i] in (ii) by G _n . Indeed, we get

  $$\displaystyle\begin{array}{rcl} E[S_{n}R_{n}(b)]& =& \frac{1} {n}\displaystyle\sum _{i=1}^{n}\Big(\eta (G_{ n}^{[i]}(b))G_{ n}^{[i]}(b) + r_{ n,1}^{[i]}(b)\Big) \\ & =& \frac{1} {n}\displaystyle\sum _{i=1}^{n}\Big(\eta \big(G_{ n}(b) + r_{n,2}^{[i]}(b)\big)\big(G_{ n}(b) + r_{n,2}^{[i]}(b)\big) + r_{ n,1}^{[i]}(b)\Big) \\ & =& \eta (G_{n}(b))G_{n}(b) + \frac{1} {n}\displaystyle\sum _{i=1}^{n}r_{ n,3}^{[i]}(b) \\ \end{array}$$

  for all b ∈ ρ _m, n , where the function

  $$r_{n,3}^{[i]} :\ \rho _{ m}(S_{n}) \cap \rho _{m}(S_{n}^{[i]}) \rightarrow \mathrm{M}_{ m}(\mathbb{C})$$

  is defined by

  $$r_{n,3}^{[i]}(b) :=\eta (G_{ n}(b))r_{n,2}^{[i]}(b)+\eta (r_{ n,2}^{[i]}(b))G_{ n}(b)+\eta (r_{n,2}^{[i]}(b))r_{ n,2}^{[i]}(b)+r_{ n,1}^{[i]}(b).$$

• (v) Combining the results from (i) and (iv), it follows

  $$bG_{n}(b) - 1 = E[S_{n}R_{n}(b)] =\eta (G_{n}(b))G_{n}(b) + \Theta _{n}(b),$$

  where we define

  $$\Theta _{n} :\ \rho _{m,n} \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b\mapsto \frac{1} {n}\displaystyle\sum _{i=1}^{n}r_{ n,3}^{[i]}(b).$$

  Due to (8), Lemmas 3.4 and 3.7 show that $\Omega _{n} \subseteq \rho _{m,n}$ and $G_{n}(b) \in \mathrm{GL}_{m}(\mathbb{C})$ for $b \in \Omega _{n}$. This gives

  $$\big(b - \Theta _{n}(b)G_{n}{(b)}^{-1}\big)G_{ n}(b) = 1 +\eta (G_{n}(b))G_{n}(b)$$

  and hence, as desired, for all $b \in \Omega _{n}$

  $$\Lambda _{n}(b)G_{n}(b) = 1 +\eta (G_{n}(b))G_{n}(b).$$

• (v) The definition of $\Omega _{n}$ gives, by Lemma 3 and by Lemma 4.1, the following estimates

  $$\|G_{n}(b)\| \leq \| R_{n}(b)\| \leq \frac{\theta } {1-\theta }\cdot \frac{1} {\|S_{n}\|} \leq \frac{\theta } {1-\theta }\cdot { \frac{1} {\|\alpha \|}^{\frac{1} {2} }} ,\qquad b \in \Omega _{n}$$

  and

  $$\|G_{n}^{[i]}(b)\| \leq \| R_{ n}^{[i]}(b)\| \leq \frac{\theta } {1-\theta }\cdot \frac{1} {\|S_{n}^{[i]}\|} \leq \frac{\theta } {1-\theta }\cdot \frac{1} {{\sqrt{1 - \frac{1} {n}}\|\alpha \|}^{\frac{1} {2} }} ,\qquad b \in \Omega _{n}.$$

  Therefore, we have for all $b \in \Omega _{n}$ by (ii)

  $$\|r_{n,1}^{[i]}(b)\| \leq \frac{1} {\sqrt{n}}\|X{_{i}\|}^{3}\|R_{ n}(b)\|\|R_{n}^{[i]}{(b)\|}^{2} \leq \frac{1} {\sqrt{n}} \frac{n} {n - 1}\Big{( \frac{\theta } {1-\theta }{\frac{1} {\|\alpha \|}^{\frac{1} {2} }} \Big)}^{3}\|X{_{ i}\|}^{3}$$

  and by (iii)

  $$\|r_{n,2}^{[i]}(b)\| \leq \frac{1} {\sqrt{n}}\|X_{i}\|\|R_{n}(b)\|\|R_{n}^{[i]}(b)\| \leq \frac{1} {\sqrt{n - 1}}\Big{( \frac{\theta } {1-\theta }{\frac{1} {\|\alpha \|}^{\frac{1} {2} }} \Big)}^{2}\|X_{ i}\|$$

  and finally by (iv)

  $$\displaystyle\begin{array}{rcl} \|r_{n,3}^{[i]}(b)\|& \leq & 2M\|G_{ n}(b)\|\|r_{n,2}^{[i]}(b)\| + M\|r_{ n,2}^{[i]}{(b)\|}^{2} +\| r_{ n,1}^{[i]}(b)\| \\ & \leq & \frac{1} {\sqrt{n - 1}}\Big{( \frac{\theta } {1-\theta }{\frac{1} {\|\alpha \|}^{\frac{1} {2} }} \Big)}^{3}\|X_{ i}\| \cdot \\ & &\bigg(2M + \frac{1} {\sqrt{n - 1}}M\Big( \frac{\theta } {1-\theta }{\frac{1} {\|\alpha \|}^{\frac{1} {2} }} \Big)\|X_{i}\| + \sqrt{ \frac{n} {n - 1}}\|X{_{i}\|}^{2}\bigg) \\ & \leq & \frac{C} {\sqrt{n}} \\ \end{array}$$

  for all $b \in \Omega _{n}$, where C > 0 is a constant, which is independent of n. Hence, it follows from (v) that

  $$\sup _{b\in \Omega _{n}}\|\Theta _{n}(b)\| \leq \frac{C} {\sqrt{n}}.$$

   □ 

The definition of $\Omega _{n}$ ensures, that

$$G := G_{s} :\ \rho _{m}(s) \rightarrow \mathrm{M}_{m}(\mathbb{C})$$

satisfies

$$bG(b) = 1 +\eta (G(b))G(b)\qquad \mbox{ for all $b \in \Omega $},$$

where

$$\Omega :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\theta \cdot \frac{1} {\|s\|},\ \|b\| \cdot \| {b}^{-1}\| < c\Big\} \supseteq \Omega _{ n}.$$

We choose

$$0 <\gamma < \frac{c - 1} {c + 1}\qquad \mbox{ and}\qquad 0 {<\theta }^{{\ast}} < (1-\gamma )\theta$$

(9)

(note, that 0 < γ < 1) and we put ${c}^{{\ast}} := c - (1 + c)\gamma$, which fulfills clearly 1 < c ^∗ < c. Since we have θ ^∗ < θ and c ^∗ < c, we see

$$\frac{\theta^{{\ast}}} {1 {-\theta }^{{\ast}}}{c}^{{\ast}} < \frac{\theta } {1-\theta }c <\sigma$$

and hence

$$ \frac{\theta^{{\ast}}} {1 {-\theta }^{{\ast}}} < \frac{\sigma } {{c}^{{\ast}}}.$$

(10)

Finally, we define

$$\kappa _{n}^{{\ast}} :{=\theta }^{{\ast}}\min \Big\{\frac{1} {\|s\|}, \frac{1} {\|S_{n}\|}, \frac{1} {\|S_{n}^{[1]}\|},\ldots , \frac{1} {\|S_{n}^{[n]}\|}\Big\}$$

and

$$\Omega _{n}^{{\ast}} :=\Big\{ b \in \mathrm{GL}_{ m}(\mathbb{C})\mid \|{b}^{-1}\| <\kappa _{ n}^{{\ast}},\ \|b\| \cdot \| {b}^{-1}\| < {c}^{{\ast}}\Big\}\subseteq \Omega _{ n}.$$

Corollary 4.4.

There exists $N \in \mathbb{N}$ such that

$$\Lambda _{n}(\Omega _{n}^{{\ast}}) \subseteq \Omega _{ n}\qquad \mbox{ for all $n \geq N$}.$$

Proof.

Since we have by Theorem 4.3

$$\sup _{b\in \Omega _{n}}\|\Theta _{n}(b)\| \leq \frac{C} {\sqrt{n}}$$

for all $2 \leq n \in \mathbb{N}$, we can choose an $N \in \mathbb{N}$ such that

$$\sup _{b\in \Omega _{n}}\|\Theta _{n}(b)\| \leq \frac{\gamma } {{c}^{{\ast}}}(1-\sigma )$$

holds for all n ≥ N. Now, we get for all $b \in \Omega _{n}^{{\ast}}$:

• (i) $\Lambda _{n}(b)$ is invertible: Since (4) gives

  $$\|G_{n}{(b)}^{-1}\| \leq \frac{1} {1-\sigma }\|b\|\qquad \mbox{ for all $b \in \Omega _{n}$},$$

  we immediately get

  $$\|\Lambda _{n}(b) - b\| \leq \| \Theta _{n}(b)\|\|G_{n}{(b)}^{-1}\| <\gamma \frac{\|b\|} {{c}^{{\ast}}} <\gamma \frac{1} {\|{b}^{-1}\|} < \frac{1} {\|{b}^{-1}\|}$$

• (ii) We have $\|\Lambda _{n}{(b)}^{-1}\| <\kappa _{n}$: Using Lemma 3.1, we get from (i) that

  $$\|\Lambda _{n}{(b)}^{-1}\| \leq \frac{1} {1-\gamma }\|{b}^{-1}\| < \frac{\kappa _{n}^{{\ast}}} {1-\gamma } <\kappa _{n}.$$

• iii) We have $\|\Lambda _{n}(b)\|\|\Lambda _{n}{(b)}^{-1}\| < c$: Using

  $$\|\Lambda _{n}(b) - b\| <\gamma \frac{\|b\|} {{c}^{{\ast}}}$$

  from (i) and

  $$\|\Lambda _{n}{(b)}^{-1}\| < \frac{1} {1-\gamma }\|{b}^{-1}\|$$

  from (ii), we get

  $$\displaystyle\begin{array}{rcl} \|\Lambda _{n}(b)\|\|\Lambda _{n}{(b)}^{-1}\|& \leq &\big(\|b\| +\| \Lambda _{ n}(b) - b\|\big)\|\Lambda _{n}{(b)}^{-1}\| \\ & <& \Big(1 + \frac{\gamma } {{c}^{{\ast}}}\Big) \frac{1} {1-\gamma }\cdot \| b\|\|{b}^{-1}\| \\ & <& \frac{{c}^{{\ast}}+\gamma } {1-\gamma } < c.\end{array}$$

Finally, this shows $\Lambda _{n}(b) \in \Omega _{n}$. □ 

Corollary 4.5.

For all n ≥ N we have

$$G_{n}(b) = G(\Lambda _{n}(b))\qquad \mbox{ for all $b \in \Omega _{n}^{{\ast}}$}.$$

Proof.

For all $n \in \mathbb{N}$ we define

$$\Omega _{n}^\prime :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|b\| < \frac{\kappa _{n}} {1-\theta }\Big\}.$$

Let n ≥ N and $b \in \Omega _{n}^{{\ast}}$ be given. We know, that

$$\Lambda _{n}(b)G(\Lambda _{n}(b)) = 1 +\eta (G(\Lambda _{n}(b)))G(\Lambda _{n}(b))$$

holds, i.e. $w = G(\Lambda _{n}(b)) \in \Omega _{n}^\prime$ is a solution of the equation

$$\Lambda _{n}(b)w = 1 +\eta (w)w,\qquad w \in \Omega _{n}^\prime .$$

Combining (8) with Lemma 4.1, we get

$$\frac{\theta } {1-\theta } <\sigma \min \Big\{ \frac{1} {c},\ \frac{\|\alpha \|} {M}\Big\} \leq \sigma \min \Big\{ \frac{1} {c},\ \frac{\|S{_{n}\|}^{2}} {M} ;\ n \in \mathbb{N}\Big\}.$$

Hence, the equation above has, by Theorem 3.9, the unique solution $w = G_{n}(b) \in \Omega _{n}^\prime$. This implies, as desired, $G_{n}(b) = G(\Lambda _{n}(b))$. □ 

Corollary 4.6.

For all n ≥ N we have

$$\|G(b) - G_{n}(b)\| \leq C^\prime \frac{1} {\sqrt{n}}\|b\|\qquad \mbox{ for all $b \in \Omega _{n}^{{\ast}}$},$$

where C′ > 0 is a constant independent of n.

Proof.

For all $b \in \Omega _{n}^{{\ast}}\subseteq \Omega _{n} \subseteq \Omega$ we have

$$\displaystyle\begin{array}{rcl} G(b) - G_{n}(b)& =& G(b) - G(\Lambda _{n}(b)) \\ & =& E\big[{(b - s)}^{-1} - {(\Lambda _{ n}(b) - s)}^{-1}\big] \\ & =& E\big[{(b - s)}^{-1}(\Lambda _{ n}(b) - b){(\Lambda _{n}(b) - s)}^{-1}\big] \\ \end{array}$$

and therefore by (4), which gives

$$\|G_{n}{(b)}^{-1}\| \leq \frac{1} {1-\sigma }\|b\|\qquad \mbox{ for all $b \in \Omega _{n}^{{\ast}}$},$$

and (since $\Lambda _{n}(b) \in \Omega _{n} \subseteq \Omega$) by (3)

$$\displaystyle\begin{array}{rcl} \|G(b) - G_{n}(b)\|& \leq & \|{(b - s)}^{-1}\| \cdot \| \Lambda _{ n}(b) - b\| \cdot \| {(\Lambda _{n}(b) - s)}^{-1}\| \\ & \leq & \Big{( \frac{\theta } {1-\theta }\cdot \frac{1} {\|s\|}\Big)}^{2} \cdot \| \Theta _{ n}(b)\| \cdot \| G_{n}{(b)}^{-1}\| \\ & \leq & C^\prime \frac{1} {\sqrt{n}}\|b\|, \\ \end{array}$$

where

$$C^\prime := \frac{C} {1-\sigma }\Big{( \frac{\theta } {1-\theta }\cdot \frac{1} {\|s\|}\Big)}^{2} > 0.$$

This proves the corollary. □ 

We recall, that the sequence $(X_{i})_{i\in \mathbb{N}}$ is bounded, which implies boundedness of the sequence $(S_{n})_{n\in \mathbb{N}}$ as well. This has the important consequence, that

$$\kappa _{n}^{{\ast}} {=\theta }^{{\ast}}\min \Big\{\frac{1} {\|s\|}, \frac{1} {\|S_{n}\|}, \frac{1} {\|S_{n}^{[1]}\|},\ldots , \frac{1} {\|S_{n}^{[n]}\|}\Big\} {\geq \kappa }^{{\ast}}$$

for some κ ^∗ > 0. If we define

$${\Omega }^{{\ast}} :=\Big\{ b \in \mathrm{GL}_{ m}(\mathbb{C})\mid \|{b}^{-1}\| {<\kappa }^{{\ast}},\ \|b\| \cdot \| {b}^{-1}\| < {c}^{{\ast}}\Big\},$$

we easily see ${\Omega }^{{\ast}}\subseteq \Omega _{n}^{{\ast}}$ for all $n \in \mathbb{N}$. Hence, by renaming ${\Omega }^{{\ast}}$ to $\Omega$ etc., we have shown our main Theorem 1.1.

We conclude this section with the following remark about the geometric structure of subsets of $\mathrm{M}_{m}(\mathbb{C})$ like $\Omega$.

Lemma 4.7.

For κ > 0 and c > 1 we consider

$$\Omega :=\Big\{ b \in \mathrm{GL}_{m}(\mathbb{C})\mid \|{b}^{-1}\| <\kappa ,\ \|b\| \cdot \| {b}^{-1}\| < c\Big\}.$$

For $\lambda ,\mu \in \mathbb{C}\setminus \{0\}$ we define

$$\Lambda (\lambda ,\mu ) := \left (\begin{array}{cccc} \lambda &0&\ldots &0\\ 0 & \mu &\ldots &0\\ \vdots & \vdots & \ddots & \vdots \\ 0&0&\ldots & \mu \end{array} \right ) \in \mathrm{GL}_{m}(\mathbb{C}).$$

If $\frac{1} {\kappa } < \vert \mu \vert$ holds, we have $\Lambda (\lambda ,\mu ) \in \Omega$ for all

$$\max \Big\{\frac{1} {\kappa } , \frac{\vert \mu \vert } {c}\Big\} < \vert \lambda \vert < c\vert \mu \vert.$$

(11)

Particularly, we have for all $\vert \lambda \vert > \frac{1} {\kappa }$, that $\lambda 1 \in \Omega$.

Proof.

Let $\mu \in \mathbb{C}\setminus \{0\}$ with $\frac{1} {\kappa } < \vert \mu \vert$ be given. For all $\lambda \in \mathbb{C}\setminus \{0\}$, which satisfy (11), we get

$$\|\Lambda {(\lambda ,\mu )}^{-1}\| =\| \Lambda {(\lambda }^{-1}{,\mu }^{-1})\| =\max \big\{ \vert \lambda {\vert }^{-1},\vert \mu {\vert }^{-1}\big\} <\kappa.$$

and

$$\displaystyle\begin{array}{rcl} \|\Lambda (\lambda ,\mu )\| \cdot \| \Lambda {(\lambda ,\mu )}^{-1}\|& =& \max \big\{\vert \lambda \vert ,\vert \mu \vert \big\}\cdot \max \big\{\vert \lambda {\vert }^{-1},\vert \mu {\vert }^{-1}\big\} \\ & =& \left \{\begin{array}{@{}l@{\quad }l@{}} \vert \mu \vert \vert \lambda {\vert }^{-1},\quad &\mbox{ if $\vert \lambda \vert < \vert \mu \vert $} \\ \vert \lambda \vert \vert \mu {\vert }^{-1},\quad &\mbox{ if $\vert \lambda \vert \geq \vert \mu \vert $} \end{array} \right. \\ & <& c, \\ \end{array}$$

which implies $\Lambda (\lambda ,\mu ) \in \Omega$. In particular, for $\lambda \in \mathbb{C}\setminus \{0\}$ with $\vert \lambda \vert > \frac{1} {\kappa }$ we see that μ = λ fulfills (11) and it follows $\lambda 1 = \Lambda (\lambda ,\lambda ) \in \Omega$. □ 

4.3 Application to Multivariate Situation

4.3.1 Multivariate Free Central Limit Theorem

Let $(x_{i}^{(k)})_{k=1}^{d}$, $i \in \mathbb{N}$, be free and identically distributed sets of d self-adjoint non-zero random variables in some non-commutative C ^∗-probability space $(\mathcal{C},\tau )$, with τ faithful, such that

$$\tau (x_{i}^{(k)}) = 0\qquad \mbox{ for $k = 1,\ldots ,d$ and all $i \in \mathbb{N}$}$$

and

$$\sup _{i\in \mathbb{N}}\max _{k=1,\ldots ,d}\|x_{i}^{(k)}\| < \infty.$$

(12)

We denote by $\Sigma = (\sigma _{k,l})_{k,l=1}^{d}$, where $\sigma _{k,l} :=\tau (x_{i}^{(k)}x_{i}^{(l)})$, their joint covariance matrix. Moreover, we put

$$S_{n}^{(k)} := \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}x_{ i}^{(k)}\qquad \mbox{ for $k = 1,\ldots ,d$ and all $n \in \mathbb{N}$}.$$

We know (cf. [11]), that $(S_{n}^{(1)},\ldots ,S_{n}^{(d)})$ converges in distribution as n → ∞ to a semicircular family $(s_{1},\ldots ,s_{d})$ of covariance $\Sigma$. For notational convenience we will assume that $s_{1},\ldots ,s_{d}$ live also in $(\mathcal{C},\tau )$; this can always be achieved by enlarging $(\mathcal{C},\tau )$.

Using Proposition 2.1 and Proposition 2.3 in [6], for each polynomial p of degree g in d non-commuting variables vanishing in 0, we can find $m \in \mathbb{N}$ and $a_{1},\ldots ,a_{d} \in \mathrm{M}_{m}(\mathbb{C})$ such that

$$\lambda 1 - p(S_{n}^{(1)},\ldots ,S_{ n}^{(d)})\quad \mbox{ and}\quad \lambda 1 - p(s_{ 1},\ldots ,s_{d})$$

are invertible in $\mathcal{C}$ if and only if

$$\Lambda (\lambda ,1) - S_{n}\quad \mbox{ and}\quad \Lambda (\lambda ,1) - s,$$

respectively, are invertible in $\mathcal{A} = \mathrm{M}_{m}(\mathcal{C})$. The matrices $\Lambda (\lambda ,1) \in \mathrm{M}_{m}(\mathbb{C})$ were defined in Lemma 4.7, and S _n and s are defined as follows:

$$S_{n} :=\displaystyle\sum _{ k=1}^{d}a_{ k} \otimes S_{n}^{(k)} \in \mathcal{A}\qquad \mbox{ for all $n \in \mathbb{N}$}$$

and

$$s :=\displaystyle\sum _{ k=1}^{d}a_{ k} \otimes s_{k} \in \mathcal{A}.$$

If we also put

$$X_{i} :=\displaystyle\sum _{ k=1}^{d}a_{ k} \otimes x_{i}^{(k)} \in \mathcal{A}\qquad \mbox{ for all $i \in \mathbb{N}$},$$

then we have

$$S_{n} = \frac{1} {\sqrt{n}}\displaystyle\sum _{i=1}^{n}X_{ i}.$$

We note, that the sequence $(X_{i})_{i\in \mathbb{N}}$ is ∗ -free with respect to the conditional expectation $E : \mathcal{A} = \mathrm{M}_{m}(\mathcal{C}) \rightarrow \mathrm{M}_{m}(\mathbb{C})$ and that all the X _i ’s have the same ∗ -distribution with respect to E and that they satisfy E[X _i ] = 0. In addition, (12) implies $\sup _{i\in \mathbb{N}}\|X_{i}\| < \infty$. Hence, the conditions of Theorem 1.1 are fulfilled. But before we apply it, we note that $(S_{n})_{n\in \mathbb{N}}$ converges in distribution (with respect to E) to s, which is an $\mathrm{M}_{m}(\mathbb{C})$-valued semicircular element with covariance mapping

$$\eta :\ \mathrm{M}_{m}(\mathbb{C}) \rightarrow \mathrm{M}_{m}(\mathbb{C}),\ b\mapsto E[sbs],$$

which is given by

$$\eta (b) = E[sbs] =\displaystyle\sum _{ k,l=1}^{d}\mathrm{id} \otimes \tau [(a_{ k} \otimes s_{k})(b \otimes 1)(a_{l} \otimes s_{l})] =\displaystyle\sum _{ k,l=1}^{d}a_{ k}ba_{l}\sigma _{k,l}.$$

Now, we get from Theorem 1.1 constants κ ^∗ > 0, c ^∗ > 0 and C′ > 0 and $N \in \mathbb{N}$ such that we have for the difference of the operator-valued Cauchy transforms

$$G_{s}(b) := E[{(b - s)}^{-1}]\quad \mbox{ and}\quad G_{ S_{n}}(b) := E[{(b - S_{n})}^{-1}]$$

the estimate

$$\|G_{s}(b) - G_{S_{n}}(b)\| \leq C^\prime \frac{1} {\sqrt{n}}\|b\|\qquad \mbox{ for all $b \in {\Omega }^{{\ast}}$ and $n \geq N$},$$

where we put

$${\Omega }^{{\ast}} :=\Big\{ b \in \mathrm{GL}_{ m}(\mathbb{C})\mid \|{b}^{-1}\| {<\kappa }^{{\ast}},\ \|b\| \cdot \| {b}^{-1}\| < {c}^{{\ast}}\Big\}.$$

Moreover, Proposition 2.3 in [6] tells us

$$\big{(\lambda 1 - p(S_{n}^{(1)},\ldots ,S_{ n}^{(d)})\big)}^{-1} = (\pi \otimes \mathrm{id}_{ \mathcal{C}})\big({(\Lambda (\lambda ,1) - S_{n})}^{-1}\big)$$

and

$$\big{(\lambda 1 - p(s_{1},\ldots ,s_{d})\big)}^{-1} = (\pi \otimes \mathrm{id}_{ \mathcal{C}^\prime })\big({(\Lambda (\lambda ,1) - s)}^{-1}\big),$$

where $\pi : \mathrm{M}_{m}(\mathbb{C}) \rightarrow \mathbb{C}$ is the mapping given by $\pi ((a_{i,j})_{i,j=1,\ldots ,m}) := a_{1,1}$. Since $\tau \circ (\pi \otimes \mathrm{id}_{\mathcal{C}}) =\pi \circ E$, this implies a direct connection between the operator-valued Cauchy transforms of S _n and s and the scalar-valued Cauchy transforms of $P_{n} := p(S_{n}^{(1)},\ldots ,S_{n}^{(d)})$ and $P := p(s_{1},\ldots ,s_{d})$, respectively. To be more precise, we get

$$G_{P_{n}}(\lambda ) :=\tau [{(\lambda -P_{n})}^{-1}] =\pi \big (G_{ S_{n}}(\Lambda (\lambda ,1))\big)$$

and

$$G_{P}(\lambda ) :=\tau [{(\lambda -P)}^{-1}] =\pi \big (G_{ s}(\Lambda (\lambda ,1))\big)$$

for all $\lambda \in \rho _{\mathcal{C}}(P_{n})$ and $\lambda \in \rho _{\mathcal{C}}(P)$, respectively.

If we choose $\mu \in \mathbb{C}$ such that $\vert \mu \vert >{ \frac{1} {\kappa }^{{\ast}}}$ holds, it follows from Lemma 4.7, that $\Lambda (\lambda ,\mu ) \in {\Omega }^{{\ast}}$ is fulfilled for all λ ∈ A(μ), where $A(\mu ) \subseteq \mathbb{C}$ denotes the open set of all $\lambda \in \mathbb{C}$ satisfying (11), i.e.

$$A(\mu ) :=\Big\{\lambda \in \mathbb{C}\mid \max \Big\{{\frac{1} {\kappa }^{{\ast}}}, \frac{\vert \mu \vert } {{c}^{{\ast}}}\Big\} < \vert \lambda \vert < {c}^{{\ast}}\vert \mu \vert \Big\}.$$

If we apply Propositions 2.1 and 2.2 in [6] to the polynomial ${\frac{1} {\mu }^{g}}p$ (which corresponds to the operators $\frac{1} {\mu } S_{n}$ and $\frac{1} {\mu } S$), we easily deduce that

$$\lambda 1 -{ \frac{1} {\mu }^{g-1}}p(S_{n}^{(1)},\ldots ,S_{ n}^{(d)})\quad \mbox{ and}\quad \lambda 1 -{ \frac{1} {\mu }^{g-1}}p(s_{1},\ldots ,s_{d})$$

are invertible in $\mathcal{C}$ if and only if

$$\Lambda (\lambda ,\mu ) - S_{n}\quad \mbox{ and}\quad \Lambda (\lambda ,\mu ) - S_{n},$$

respectively, are invertible in $\mathcal{A}$. Moreover, we have

$$\mu^{g-1}G_{ P_{n}}{(\lambda \mu }^{g-1}) =\pi \big (G_{ S_{n}}(\Lambda (\lambda ,\mu ))\big)$$

and

$$\mu^{g-1}G_{ P}{(\lambda \mu }^{g-1}) =\pi \big (G_{ s}(\Lambda (\lambda ,\mu ))\big)$$

for all $\lambda \in \rho _{\mathcal{C}}({ \frac{1} {\mu }^{g-1}} P_{n})$ and $\lambda \in \rho _{\mathcal{C}}({ \frac{1} {\mu }^{g-1}} P)$, respectively.

Particularly, for all λ ∈ A(μ) we get $\Lambda (\lambda ,\mu ) \in {\Omega }^{{\ast}}$ and hence $\lambda \in \rho _{\mathcal{C}}({ \frac{1} {\mu }^{g-1}} P_{n}) \cap \rho _{\mathcal{C}}({ \frac{1} {\mu }^{g-1}} P)$ for all n ≥ N. Therefore, Theorem 1.1 implies

$$\displaystyle\begin{array}{rcl} \vert \mu {\vert }^{g-1}\vert G_{ P}{(\lambda \mu }^{g-1}) - G_{ P_{n}}{(\lambda \mu }^{g-1})\vert & =& \big\vert \pi \big(G_{ s}(\Lambda (\lambda ,\mu )) - G_{S_{n}}(\Lambda (\lambda ,\mu ))\big)\big\vert \\ &\leq & \big\|G_{s}(\Lambda (\lambda ,\mu )) - G_{S_{n}}(\Lambda (\lambda ,\mu ))\big\| \\ & \leq & C^\prime \frac{1} {\sqrt{n}}\|\Lambda (\lambda ,\mu )\| \\ & \leq & C^\prime \frac{1} {\sqrt{n}}\max \{\vert \lambda \vert ,\vert \mu \vert \} \\ &\leq & C^\prime {c}^{{\ast}}\vert \lambda \vert \frac{1} {\sqrt{n}} \\ \end{array}$$

and hence

$$\vert G_{P}{(\lambda \mu }^{g-1}) - G_{ P_{n}}{(\lambda \mu }^{g-1})\vert \leq C^\prime {c}^{{\ast}} \frac{1} {\sqrt{n}}{\vert \lambda \mu }^{g-1}\vert.$$

This means, that

$$\vert G_{P}(z) - G_{P_{n}}(z)\vert \leq C^\prime {c}^{{\ast}} \frac{1} {\sqrt{n}}\vert z\vert $$

holds for all $z \in \mathbb{C}$ with ${ \frac{z} {\mu }^{g-1}} \in A(\mu )$ and all n ≥ N. By definition of A(μ), we particularly get

$$\vert G_{P}(z) - G_{P_{n}}(z)\vert \leq C \frac{1} {\sqrt{n}}\qquad \mbox{ for all $ \frac{1} {{c}^{{\ast}}}\vert \mu {\vert }^{g} < \vert z\vert < {c}^{{\ast}}\vert \mu {\vert }^{g}$ and $n \geq N$},$$

where we put $C := C^\prime {({c}^{{\ast}})}^{2}\vert \mu {\vert }^{g} > 0$. Since $z\mapsto G_{P}(z) - G_{P_{n}}(z)$ is holomorphic on $\{z \in \mathbb{C}\mid \vert z\vert > R\}$ for $R := \frac{1} {{c}^{{\ast}}}\vert \mu {\vert }^{g} > 0$ and extends holomorphically to ∞, the maximum modulus principle gives

$$\vert G_{P}(z) - G_{P_{n}}(z)\vert \leq C \frac{1} {\sqrt{n}}\qquad \mbox{ for all $\vert z\vert > R$ and $n \geq N$}.$$

This shows Theorem 1.2 in the case of a polynomial p vanishing in 0. For a general polynomial p, we consider the polynomial $\tilde{p} = p - p_{0}$ with $p_{0} := p(0,\ldots ,0)$, which leads to the operators $\tilde{P} = P - p_{0}1$ and $\tilde{P}_{n} = P_{n} - p_{0}1$. Since we can apply the result above to $\tilde{p}$ and since the Cauchy transforms G _P and $G_{P_{n}}$ are just translations of $G_{\tilde{P}}$ and $G_{\tilde{P}_{n}}$, respectively, the general statement follows easily.

4.3.2 Estimates in Terms of the Kolmogorov Distance

In the classical case, estimates between scalar-valued Cauchy transforms can be established (for self-adjoint operators) in all of the upper complex plane and lead then to estimates in terms of the Kolmogorov distance. In the case treated above, we have a statement about the behavior of the difference between two Cauchy transforms only near infinity. Even in the case, where our operators are self-adjoint, we still have to transport estimates from infinity to the real line, and hence we can not apply the results of Bai [1] directly. A partial solution to this problem was given in the appendix of [14] with the following theorem, formulated in terms of probability measures instead of operators. There we use the notation G _μ for the Cauchy transform of the measure μ, and put

$$D_{R}^{+} :=\{ z \in \mathbb{C}\mid \mathrm{Im}(z) > 0,\ \vert z\vert > R\}.$$

Theorem 4.8.

Let μ be a probability measure with compact support contained in an interval [−A,A] such that the cumulative distribution function $\mathcal{F}_{\mu }$ satisfies

$$\vert \mathcal{F}_{\mu }(x + t) -\mathcal{F}_{\mu }(x)\vert \leq \rho \vert t\vert \qquad \mbox{ for all $x,t \in \mathbb{R}$}$$

for some constant ρ > 0. Then for all R > 0 and β ∈ (0,1) we can find $\Theta > 0$ and m₀ > 0 such that for any probability measure ν with compact support contained in [−A,A], which satisfies

$$\sup _{z\in D_{R}^{+}}\vert G_{\mu }(z) - G_{\nu }(z)\vert \leq {e}^{-m}$$

for some m > m ₀ , the Kolmogorov distance $\Delta (\mu ,\nu ) :=\sup _{x\in \mathbb{R}}\vert \mathcal{F}_{\mu }(x) -\mathcal{F}_{\nu }(x)\vert$ fulfills

$$\Delta (\mu ,\nu ) \leq \Theta \frac{1} {{m}^{\beta }}.$$

Obviously, this leads to the following questions: First, the stated estimate for the speed of convergence in terms of the Kolmogorov distance is far from the expected one. We hope to improve this result in a future work. Furthermore, in order to apply this theorem, we have to ensure that $p(s_{1},\ldots ,s_{d})$ has a continuous density. As mentioned in the introduction, it is a still unsolved problem, whether this is always true for any self-adjoint polynomials p.