Asymptotic Spectra of Matrix-Valued Functions of Independent Random Matrices and Free Probability

Introduction

One of the main questions studied in Random Matrix Theory is the asymptotic universality, meaning the dependence on a few global characteristics of the distribution of the matrix entries, of the distribution of spectra of random matrices when their dimension goes to infinity. This holds for the spectra of Hermitian random matrices with independent entries (up to symmetry), first proved by Wigner in 1955 . Another well studied case is that of sample covariance matrices (i.e. $\mathbf{W}=\mathbf{X}\mathbf{X}^{*}$, where $\mathbf{X}$ is a matrix with independent entries), first studied in by Marchenko–Pastur. The spectrum of non Hermitian random matrices with independent identically distributed entries is universal as well. The limiting complex spectrum of this Ginibre–Girko Ensemble is the circular law (i.e. the uniform distribution on the unit circle in the complex plane). The universality here was first proved in by Girko. In the last years different models of random matrices which were derived from Wigner and Ginibre–Girko matrices were studied. For instance, in , the universality of the singular value distribution of powers of Ginibre–Girko matrices was shown. In and the universality of the spectrum of products of independent random matrices from the Ginibre–Girko Ensemble was proved. Moreover, more recently, the local properties of the spectrum have also been investigated in the Gaussian case; see e.g. and .

In this paper we describe a general approach to prove the universality of singular value and eigenvalue distributions of matrix-valued functions of independent random matrices. More precisely, we consider random matrices of the form

Furthermore, we introduce a general approach to identify the limiting eigenvalue distribution of the (square) matrix $\mathbf{F}$. Our main results here show how to derive the density of the limiting eigenvalue distribution of the matrix $\mathbf{F}$ from (the $S$-transform of) its limiting singular value distribution. This derivation can be divided into two major steps:

In a first step, we derive equations for the Stieltjes transforms $g(z,\alpha)$ of the (symmetrized) singular value distributions of the shifted matrices $\mathbf{F}-\alpha\mathbf{I}$ via the $S$-transform $S(z)$ of the (symmetrized) singular value distribution of the unshifted matrix $\mathbf{F}$. The key system of equations here reads

where $w(z,\alpha)$ is an unknown auxiliary function and $\widetilde{R}_{\alpha}(z)$ is a known function. To derive this system of equations, we use the asymptotic freeness of the matrices

as well as the calculus for $R$-transforms and $S$-transforms. Furthermore, we show that it is possible take the limit $z\to 0$ in (1). Since we are working in a quite general framework, the investigation of the existence of this limit as well as its analytic properties require some work.

In a second step, we identify the density $f$ of the limiting eigenvalue distribution of the random matrix $\mathbf{F}$ using logarithmic potential theory. The main observation here is that the function $\psi(\alpha):=-w(0,\alpha)g(0,\alpha)$ is closely related to the partial derivatives of the logarithmic potential of the limiting eigenvalue distribution. Thus, under regularity assumptions, we obtain the relation

where $u$ and $v$ denote the real and imaginary part of $\alpha$, respectively.

Let us emphasize that this identification of the limiting eigenvalue distribution is quite general. In principle, we only need the $S$-transform of the limiting singular value distribution and the asymptotic freeness of the matrices in (1.2).

In Section 8 we give several examples for applications of our main universality results (Theorems 3.2 and 4.4). The guiding principle here is (i) to establish universality and (ii) to compute the limits in the Gaussian case, using tools from free probability theory. Here we focus on a special class of matrix-valued functions, namely products of matrices or powers and inverses thereof. Although our framework should, in principle, cover more general functions as well, products of independent matrices represent a convenient class of examples in which the assumptions of our main results can be checked. For instance, the conditions $(C0)$, $(C1)$, $(C2)$ on the large and small singular values can be deduced from existing results by Tao and Vu and Götze and Tikhomirov , here. Moreover, once universality is proved, it suffices to identify the limiting eigenvalue and singular value distributions in the Gaussian case. But if the random matrices $\mathbf{X}^{(1)},\ldots,\mathbf{X}^{(m)}$ have independent standard Gaussian entries, their distributions are invariant under rotations, and the $S$-transforms of the limiting singular value distributions of their products are readily obtained using tools from free probability theory, see e.g. Voiculescu or Hiai and Petz . From here it is possible to obtain the limiting singular value distributions and, as we have seen, the limiting eigenvalue distributions.

Our examples illustrate that our main results provide a unifying framework to derive old and new results for products of independent random matrices. In particular, we determine the limiting singular value and eigenvalue distributions for products of independent random matrices from the so-called spherical ensemble (see e.g. ), i.e. for products of the form $\mathbf{X}^{(1)}(\mathbf{X}^{(2)})^{-1}\cdots\mathbf{X}^{(2m-1)}(\mathbf{X}^{(2m)})^{-1}$, where $\mathbf{X}^{(1)},\ldots,\mathbf{X}^{(2m)}$ are independent Girko–Ginibre matrices.

General Framework

We now introduce our main assumptions and notation. Generalizations and specializations will be indicated at the beginnings of later sections.

In order to study the spectral asymptotics of sequences of such matrix tuples we shall make a so-called dimension shape assumption, meaning that $n_{q}=n_{q}(n)$, and that for any $q=1,\ldots,m$,

Let $\mathbf{X}=(\mathbf{X}^{(1)},\ldots,\mathbf{X}^{(m)})$ be an $m$-tuple of independent random matrices of dimensions $n_{0}\times n_{1},\ldots,n_{m-1}\times n_{m}$, respectively, with independent entries. More precisely, we assume that

where the $X^{(q)}_{jk}$ are independent complex random variables such that for all $q=1,\ldots,m$ and $j=1,\ldots,n_{q-1};\,k=1,\ldots,n_{q}$, we have $\mathbf{E}\,X_{jk}^{(q)}=0$ and $\mathbf{E}\,|X_{jk}^{(q)}|^{2}=1$.

Furthermore, let $\mathbf{Y}=(\mathbf{Y}^{(1)},\ldots,\mathbf{Y}^{(m)})$ be an $m$-tuple of independent random matrices of dimensions $n_{0}\times n_{1},\ldots,n_{m-1}\times n_{m}$, respectively, with independent Gaussian entries. More precisely, we assume that

In particular, $\mathbf{E}\,Y_{jk}^{(q)}=0$ and $\mathbf{E}\,|Y_{jk}^{(q)}|^{2}=1$.

In Section 8, when we determine the limiting singular value and eigenvalue distributions in the Gaussian case, we will impose the stronger assumption that the $Y^{(q)}_{jk}$ are standard real or complex Gaussian random variables. By Eq. (2), this entails some restrictions on the second moments of the $X^{(q)}_{jk}$.

We shall also assume that the random matrices $\mathbf{X}^{(1)},\ldots,\mathbf{X}^{(m)}$ and $\mathbf{Y}^{(1)},\ldots,\mathbf{Y}^{(m)}$ are defined on the same probability space and that $\mathbf{Y}^{(1)},\ldots,\mathbf{Y}^{(m)}$ are independent of $\mathbf{X}^{(1)},\ldots,\mathbf{X}^{(m)}$. Finally, for any $m$-tuple $\mathbf{Z}=(\mathbf{Z}^{(1)},\ldots,\mathbf{Z}^{(m)})$ in $\mathcal{M}_{n_{0}\times n_{1}}\times\cdots\times\mathcal{M}_{n_{m-1}\times n_{m}}$, we set

Note that since we are interested in asymptotic singular value and eigenvalue distributions, we are actually dealing with sequences of matrix tuples of increasing dimension. However, the dependence on $n$ is usually suppressed in our notation.

Throughout this paper, we use the following notation. For a matrix $\mathbf{A}=(a_{jk})$ $\in\mathcal{M}_{n\times p}$, we write $\|\mathbf{A}\|$ for the operator norm of $\mathbf{A}$ and $\|\mathbf{A}\|_{2}:=(\sum_{j=1}^{n}\sum_{k=1}^{p}|a_{jk}|^{2})^{1/2}$ for the Frobenius norm of $\mathbf{A}$. The singular values of $\mathbf{A}$ are the square-roots of the eigenvalues of the $n\times n$ matrix $\mathbf{A}\mathbf{A}^{*}$. Finally, unless otherwise indicated, $C$ and $c$ denote sufficiently large and small positive constants, respectively, which may change from step to step.

Universality of Singular Value Distributions of Functions of Independent Random Matrices

We shall assume that the random variables $X_{jk}^{(q)}$, for $q=1,\ldots,m$, $j=1,\ldots,n_{q-1};\,k=1,\ldots,n_{q}$, satisfy the following Lindeberg condition, i. e.

We now define truncated matrices. Note that by (3.1) there exists a sequence $(\tau_{n})$ such that

Clearly, we may additionally require that $\tau_{n}\geq n^{-1/3}$ for all $n$. We fix such a sequence and consider the matrix tuple $\widehat{\mathbf{X}}=(\widehat{\mathbf{X}}^{(1)},\ldots,\widehat{\mathbf{X}}^{(m)})$ consisting of the matrices $\widehat{\mathbf{X}}^{(q)}=(\tfrac{1}{\sqrt{n_{q}}}\widehat{X}_{jk}^{(q)})$, $q=1,\ldots,m$, where

Let $\mathbf{B}$ be a non-random matrix of order $n\times p$, let $\mathbf{F}_{\mathbf{X}}$ and $\mathbf{F}_{\widehat{\mathbf{X}}}$ be defined as in (2.3), and let $s_{1}({\mathbf{X}})\geq\ldots\geq s_{n}({\mathbf{X}})$ and $s_{1}({\widehat{\mathbf{X}}})\geq\ldots\geq s_{n}({\widehat{\mathbf{X}}})$ denote the singular values of the matrices $\mathbf{F}_{\mathbf{X}}+\mathbf{B}$ and $\mathbf{F}_{\widehat{\mathbf{X}}}+\mathbf{B}$, respectively. Let $\mathcal{F}_{\mathbf{X}}(x)$ (resp. $\mathcal{F}_{\widehat{\mathbf{X}}}(x)$) denote the empirical distribution function of the squared singular values of the matrix $\mathbf{F}_{\mathbf{X}}+\mathbf{B}$ (resp. $\mathbf{F}_{\widehat{\mathbf{X}}}+\mathbf{B}$), i.e.

The corresponding Stieltjes transforms of these empirical distributions are denoted by $m_{\mathbf{X}}(z)$ and $m_{\widehat{\mathbf{X}}}(z)$, i.e.

Assume that the conditions (3.1) and (3.2) hold. Then

By the rank inequality of Bai, see , Theorem A.44, we have

Inequalities (3.4)–(3) and assumption (3.3) together complete the proof of the Lemma. ∎

Let $\mathbf{Y}=(\mathbf{Y}^{(1)},\ldots,\mathbf{Y}^{(m)})$ be an $m$-tuple of independent random matrices with independent Gaussian entries as in Section 1, and let $\widehat{\mathbf{Y}}=(\widehat{\mathbf{Y}}^{(1)},\ldots,\widehat{\mathbf{Y}}^{(m)})$ denote the $m$-tuple consisting of the matrices $\widehat{\mathbf{Y}}^{(q)}=(\tfrac{1}{\sqrt{n_{q}}}\widehat{Y}^{(q)}_{jk})$, where

for $q=1,\ldots,m$ and $j=1,\ldots,n_{q-1};\,k=1,\ldots,n_{q}$. Set

Then, using the relation $\tau_{n}\geq n^{-1/3}$ and the special properties of the Gaussian distribution, it is easy to check that we also have

Furthermore, note that for the truncated random variables, the moment identities (2) need not hold anymore. However, we have the relations

as well as the analogous relations for the r.v.’s $\widehat{Y}_{jk}^{(q)}$, which imply that

For the rest of this section, we use the following notation. For any matrix tuple $\mathbf{X}=(\mathbf{X}^{(1)},\ldots,\mathbf{X}^{(m)})$ and any $n\times p$ matrix $\mathbf{B}$, we introduce the matrix $\mathbf{F}_{\mathbf{X}}$ as in (2.3), the Hermitian matrix

as well as the corresponding resolvent matrix

Furthermore, let $s_{1}({\mathbf{X}})\geq\ldots\geq s_{n}({\mathbf{X}})$ denote the singular values of the matrix $\mathbf{F}_{{\mathbf{X}}}+\mathbf{B}$. Note that, apart from a fixed number of zero eigenvalues, the eigenvalues of the matrix $\mathbf{V}_{\mathbf{X}}$ are given by $\pm s_{1}({\mathbf{X}}),\ldots\,\pm s_{n}({\mathbf{X}})$. The corresponding Stieltjes transform will be denoted by

For $0\leq\varphi\leq\frac{\pi}{2}$ and $q=1,\ldots,m$, let

and $\mathbf{Z}(\varphi):=(\mathbf{Z}^{(1)}(\varphi),\ldots,\mathbf{Z}^{(m)}(\varphi))$. For abbreviation, we shall write $\mathbf{F}(\varphi)$, $\mathbf{V}(\varphi)$, $\mathbf{R}(\varphi)$, and $m_{n}(z,\varphi)$ instead of $\mathbf{F}_{\mathbf{Z}(\varphi)}$, $\mathbf{V}_{\mathbf{Z}(\varphi)}$, $\mathbf{R}_{\mathbf{Z}(\varphi)}$, and $m_{n}(z,\mathbf{Z}(\varphi))$. With this notation we have $\mathbf{F}_{\widehat{\mathbf{X}}}=\mathbf{F}(0)$, $\mathbf{F}_{\widehat{\mathbf{Y}}}=\mathbf{F}(\frac{\pi}{2})$, $m_{n}(z,\widehat{\mathbf{X}})=m_{n}(z,0)$ and $m_{n}(z,\widehat{\mathbf{Y}})=m_{n}(z,\frac{\pi}{2})$. Also, we may write

The representation of type (3.17) has been used for sums of random variables, for instance, in (second relation on page 367). For random matrices (3.17) has been used, for example, by Pastur and Lytova in (see Equation (60)).

Let $g_{jk}^{(q)}(\theta)$ denote the function obtained from $g_{jk}^{(q)}$ by replacing the indeterminate $Z_{jk}^{(q)}$ with $\theta Z_{jk}^{(q)}$.

Assume that the Lindeberg condition (3.1) and the rank condition (3.2) hold. Furthermore suppose that there exist constants $A_{0}>0$, $A_{1}>0$ and $A_{2}>0$ such that for any random variable $\theta$ which is uniformly distributed on the interval $$and independent of the r.v.’s$X_{jk}^{(q)}$and$Y_{jk}^{(q)}$, the following conditions hold:

Remark. It follows from the conclusion of the theorem and basic properties of the Stieltjes transform that if the singular value distributions of the matrices $\mathbf{F}_{\mathbf{Y}}+\mathbf{B}$ are weakly convergent in probability to some limit $\nu$, then so are the singular value distributions of the matrices $\mathbf{F}_{\mathbf{X}}+\mathbf{B}$. In this sense Theorem 3.2 proves the universality of singular value distributions.

By Lemma 3.1 and the subsequent remark, it is sufficient to prove the claim with $m_{n}(z,\widehat{\mathbf{X}})$ and $m_{n}(z,\widehat{\mathbf{Y}})$ instead of $m_{n}(z,\mathbf{X})$ and $m_{n}(z,\mathbf{Y})$. Furthermore, according to Lemma A.1 in the Appendix, it is enough to prove that

where $\theta$ is a random variable which is uniformly distributed on the unit interval, we get

Here $\mathbf{E}\,_{\theta}$ denotes the expectation with respect to the r.v. $\theta$ conditioning on all other r.v.’s. Inserting (3) into (3), we get

and $\Sigma_{7},\ldots,\Sigma_{12}$ denote similar terms coming from the second line in (3). Since $\Sigma_{7},\ldots,\Sigma_{12}$ can be treated in the same way as $\Sigma_{1},\ldots,\Sigma_{6}$, we provide the details for the latter only.

Since $g_{jk}^{(q)}(0,0)$ and $X_{jk}^{(q)}$, $Y_{jk}^{(q)}$ are independent, it follows from (3.13) that

Using again that the random variables $\frac{\partial g_{jk}^{(q)}}{\partial Z_{jk}^{(q)}}(0,0)$ and $X_{jk}^{(q)}$, $Y_{jk}^{(q)}$ are independent, we get

By (3.14), the last inequality implies that

Similarly, using (3.9) and (3.16), we get

Combining the preceding estimates, we obtain (3.26). Thus, Theorem 3.2 is proved. ∎

Universality of Eigenvalue Distributions of Functions of Independent Random Matrices

We now turn to the eigenvalue distribution of functions of independent random matrices. We use the assumptions and the notation from Section 1, but throughout this section we assume additionally that $n=p$, so that ${\mathbf{F}}_{\mathbf{X}}$ and ${\mathbf{F}}_{\mathbf{Y}}$ are square matrices.

Let $\mu$ a probability measure on the complex plane with compact support. Define the logarithmic potential of the measure $\mu$ as

Let $\mu_{\mathbf{X}}$ (resp. $\mu_{\mathbf{Y}}$) denote the empirical spectral measure of the matrix $\mathbf{F}_{\mathbf{X}}$ (resp. $\mathbf{F}_{\mathbf{Y}}$), i.e. $\mu_{\mathbf{X}}$ (resp. $\mu_{\mathbf{Y}}$) is the uniform distribution on the eigenvalues $\{\lambda_{1}(\mathbf{X}),\ldots,\lambda_{n}(\mathbf{X})\}$ (resp. $\{\lambda_{1}(\mathbf{Y}),\ldots\lambda_{n}(\mathbf{Y})\}$) of the matrix $\mathbf{F}_{\mathbf{X}}$ (resp. $\mathbf{F}_{\mathbf{Y}}$). Then

If there exists some $p>0$ such that the quantity

is bounded in probability as $n\to\infty$, we say that the matrices $\mathbf{F}_{\mathbf{X}}$ satisfy condition $(C0)$.

holds, we say that the matrices $\mathbf{F}_{\mathbf{X}}$ satisfy condition $(C1)$.

with $n_{1}=[n-n\delta_{n}]+1$, $n_{2}=[n-n^{\gamma}]$, we say that the matrices $\mathbf{F}_{\mathbf{X}}$ satisfy condition $(C2)$.

We now prove the universality of eigenvalue distributions.

for any bounded continuous function $f$ and any $\varepsilon>0$.

The proof of Theorem 4.4 is based on the “replacement principle” by Tao and Vu (see , Theorem 2.1) which builds upon a sort of inversion formula for the logarithmic potential that goes back to Girko and that was also investigated by Bai , .

Note that by condition (C1) the determinants in (4.3) are not zero with probability $1-o(1)$.

Let $L(F,G)$ denote the Lévy distance between two distribution functions $F$ and $G$. Recall that

Let $\mathcal{G}_{\mathbf{X}}(x,\alpha)$ and $\mathcal{G}_{\mathbf{Y}}(x,\alpha)$ denote the distribution functions of the singular values of the matrices $\mathbf{F}_{\mathbf{X}}-\alpha\mathbf{I}$ and $\mathbf{F}_{\mathbf{Y}}-\alpha\mathbf{I}$, respectively. Let $\varkappa_{n}=L(\mathcal{G}_{\mathbf{X}}(\cdot,\alpha),\mathcal{G}_{\mathbf{Y}}(\cdot,\alpha))$. According to Theorem 3.2 (with $\mathbf{B}=\alpha\mathbf{I}$), we have

Note that by (4.4) and Markov’s inequality, we have

Furthermore, put $\eta_{n}=\max\{(\mathbf{E}\,\varkappa_{n})^{\frac{1}{3}},(\log n)^{-1}\}$, and introduce the event

Let $\delta_{n}:=1/|\log 2\eta_{n}|\to 0$, let $n_{1}=[n-n\delta_{n}]+1$ and $n_{2}=[n-n^{\gamma}]$ be defined as in condition (C2), and introduce the event

Note that on the set $\{s_{n_{1}}(\mathbf{F}_{\mathbf{X}}-\alpha\mathbf{I})<2\eta_{n}\}$, we have, for large enough $n$,

by condition (C2). Since a similar estimate holds with $\mathbf{F}_{\mathbf{Y}}$ instead of $\mathbf{F}_{\mathbf{X}}$, it follows that

Furthermore, on the set $\Omega_{3}$ we have, for any $a\in[\eta_{n},2\eta_{n}]$,

Now fix $\varepsilon>0$. The preceding inequality implies that

By condition $(C1)$ and inequalities (4.5) – (4.7),

By definition of $\Omega_{0}$ and by condition $(C2)$, we have, for $n\to\infty$,

Moreover, using that for any $p>0$, the function $x^{-p}\log x$ is decreasing in $x$ for $x\geq\text{\rm e}^{1/p}$, we get, for large enough $n$,

By the inequality $s_{k}(\mathbf{F}-\alpha\mathbf{I})\leq s_{k}(\mathbf{F})+|\alpha|$ and condition $(C0)$, this quantity converges to zero in probability.

Thus, it remains to bound the last but one summand in (4). Recall that $a\in[\eta_{n},2\eta_{n}]$. Integrating by parts, we have

Recall that we need to bound this expression for $\omega\in\Omega_{2}\cap\Omega_{1}$ only and that $\varkappa_{n}\leq\eta_{n}$ for such $\omega$. By Chebyshev’s inequality, we have

for any $M>0$. It therefore follows from condition $(C0)$ that the second term on the r.h.s. of (4) converges to zero in probability. Furthermore, by the definition of $\varkappa_{n}$, we have the following bound

Thus, for $\omega\in\Omega_{2}\cap\Omega_{1}$ we may find $a\in[\eta_{n},2\eta_{n}]$ such that

Because $\eta_{n}^{\frac{1}{2}}|\log\eta_{n}|\to 0$ as $n\to\infty$, it follows that the first term on the r.h.s of (4) converges to zero in probability. Finally, using inequality (4.12) again, we obtain

It is straightforward to check that for any $0<\varepsilon<a<b$ and any distribution function $F(x)$ the following inequality

Therefore, for $\omega\in\Omega_{2}\cap\Omega_{1}$ we obtain, for large enough $n$,

We may now apply the “replacement principle” by Tao and Vu; see , Theorem 2.1. Note that this theorem is based on two assumptions (i) and (ii). Assumption (ii) is just a reformulation of relation (4.3). Assumption (i) is only needed to show that the probability measures $\mu_{\mathbf{X}}$ and $\mu_{\mathbf{Y}}$ are tight in probability (see Equations (3.3) and (3.4) in ), and may be replaced with our assumption ($C0$). It therefore follows that $\mu_{\mathbf{X}}-\mu_{\mathbf{Y}}$ converges weakly to zero in probability, i.e. for any bounded continuous function $f$ and any $\varepsilon>0$, we have

Asymptotic Freeness of Random Matrices

In this section we consider the asymptotic freeness of random matrices with special structure. Before that, we recall the definition of Voiculescu’s asymptotic freeness as well as some basic notions from free probability theory. See also the survey by Speicher and the lecture notes by Voiculescu .

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space.

$1)$ Let $(\mathcal{A}_{i})_{i\in I}$ be a family of unital sub-algebras of $\mathcal{A}$. The sub-algebras $\mathcal{A}_{i}$ are called free or freely independent, if, for any positive integer $k$, $\varphi(a_{1}\cdots a_{k})=0$ whenever the following set of conditions holds: $a_{j}\in\mathcal{A}_{i(j)}$ (with $i(j)\in I$) for all $j=1,\ldots,k$, $\varphi(a_{j})=0$ for all $j=1,\ldots,k$, and neighbouring elements are from taken different sub-algebras, i.e. $i(1)\neq i(2),\,i(2)\neq i(3),\,\ldots,\,i(k-1)\neq i(k)$.

2) Let $(\mathcal{A}_{i}^{\prime})_{i\in I}$ be a family of subsets of $\mathcal{A}$. The subsets $\mathcal{A}_{i}^{\prime}$ are called free or freely independent, if their generated unital sub-algebras are free, i.e. if $(\mathcal{A}_{i})_{i\in I}$ are free, where, for each $i\in I$, $\mathcal{A}_{i}$ is the smallest unital sub-algebra of $\mathcal{A}$ which contains $\mathcal{A}_{i}^{\prime}$.

3) Let $(a_{i})_{i\in I}$ be a family of elements from $\mathcal{A}$. The elements $a_{i}$ are called free or freely independent, if the subsets $\{a_{i}\}$ are free.

Consider two random variables $a$ and $b$ which are free. Then the distributions of $a+b$ and $ab$ (in the sense of linear functionals) depend only on the distributions of $a$ and $b$ (see e.g. , Chapter 2), and we can make the following definition:

For free random variables $a$ and $b$, the distributions of $a+b$ and $ab$ are called the free additive convolution and the free multiplicative convolution of $\mu_{a}$ and $\mu_{b}$ and are denoted by

where $G_{a}^{-1}(z)$ denotes the inverse of $G_{a}(z)$ w.r.t. composition of functions. Moreover, when $\varphi(a)\neq 0$, define the $S$-transform of $a$ by

Then, for free random variables $a$ and $b$, we have

and, when $\phi(a)\neq 0$ and $\phi(b)\neq 0$,

where ${\widetilde{R}}_{a}^{-1}(z)$ denotes the inverse of ${\widetilde{R}}_{a}(z)$ w.r.t. composition of functions. See for instance Nica , Equation (21) in Chapter 13. For clarity, let us emphasize that we call $\widetilde{R}_{a}(z)$ what is called $R_{a}(z)$ in . Furthermore, let us note that the argument in requires that $\varphi(a)\neq 0$. However, when $\varphi(a)=0$ and $\varphi(a^{2})\neq 0$, one can use similar arguments as in Rao and Speicher to show that, similarly as for the $S$-transform, there exist two branches of ${\widetilde{R}}_{a}^{-1}(z)$ and that (5.5) continues to hold with an appropriate choice of these branches. We shall always take the branches such that

The above transforms also have extensions to unbounded probability measures. Let us provide the details for the $S$-transform; cf. Section 6 in . For a probability measure $\nu$ on $(0,\infty)$, define the function

where $\psi_{\nu}^{-1}$ denotes the inverse of $\psi_{\nu}$.

and the families $\{a_{i}:i\in I_{j}\}$, $j=1,\ldots,l$, are free.

Here $\mathbf{I}_{n}$ denotes the identity matrix of dimension $n\times n$.

This means that relation (5.7) holds if at least one of the $l_{1},l_{2},\ldots,l_{k}$ is even. Hence, suppose that $l_{1},\ldots,l_{k}$ are all odd. In this case we may reduce relation (5.7) to

In order to complete the proof, we proceed similarly as in Hiai and Petz . Using the bi-unitary invariance and the singular value decomposition of the matrix $\mathbf{Y}_{n}$, we may represent the matrix $\mathbf{Y}_{n}$ as $\mathbf{U}_{n}\mathbf{\Delta}_{n}\mathbf{V}_{n}^{*}$, where $\mathbf{U}_{n}$, $\mathbf{\Delta}_{n}$, $\mathbf{V}_{n}$ are independent, $\mathbf{U}_{n}$ and $\mathbf{V}_{n}$ are random unitary matrices (with Haar distribution), and $\mathbf{\Delta}_{n}$ is a random diagonal matrix whose diagonal elements are the singular values of $\mathbf{Y}_{n}$, but with random signs (chosen uniformly at random and independently from everything else). Note that

Thus, the non-zero $n\times n$ blocks in the matrix $\widetilde{\mathbf{A}_{n}^{j_{1}}}\mathbf{J}(\alpha)\cdots\widetilde{\mathbf{A}_{n}^{j_{k}}}\mathbf{J}(\alpha)$ in (5.21) are products of the matrices $\mathbf{U}_{n}(\mathbf{\Delta}_{n}^{2p}-\int x^{2p}d\mu\mathbf{I})\mathbf{U}_{n}^{*}$, $\mathbf{V}_{n}(\mathbf{\Delta}_{n}^{2p}-\int x^{2p}d\mu\mathbf{I})\mathbf{V}_{n}^{*}$, $\mathbf{U}_{n}\mathbf{\Delta}_{n}^{2p+1}\mathbf{V}_{n}^{*}$, $\mathbf{V}_{n}\mathbf{\Delta}_{n}^{2p+1}\mathbf{U}_{n}^{*}$, as well as certain powers of $\alpha$ and $\overline{\alpha}$, such that each $\mathbf{U}_{n}^{*}$ is followed by a $\mathbf{V}_{n}$, and each $\mathbf{V}_{n}^{*}$ is followed by a $\mathbf{U}_{n}$.

But this implies that the limit in (5.21) is equal to zero. This completes the proof of the asymptotic freeness of $\mathbf{A}_{n}$ and $\mathbf{B}_{n}$. ∎

The notion of bi-unitary invariance is relevant for computing the limiting spectral distributions for random matrices with i.i.d. standard complex Gaussian entries. To compute the limiting distributions for random matrices with i.i.d. standard real Gaussian entries we need the notion of bi-orthogonal invariance. According to a side-remark in Hiai and Petz , the results for this case are analogous.

Moreover, using bi-orthogonal invariance, it is even possible to treat random matrices with i.i.d. entries with a common bivariate real Gaussian distribution. (Thus, we may allow for correlations between the real and imaginary parts, for example.) Indeed, suppose that the matrices $\mathbf{Y}^{(1)},\ldots,\mathbf{Y}^{(q)}$ have i.i.d. Gaussian entries such that $\mathbf{E}\,Y_{jk}^{(q)}=0$, $\mathbf{E}\,|Y_{jk}^{(q)}|^{2}=1$ and

Since the matrices $\tau_{1}\mathbf{Y}^{\prime}_{j}+i\tau_{2}\mathbf{Y}^{\prime\prime}_{j}$ have independent complex entries with mean and variance $1$, their limiting singular value distribution is well-known by the Marchenko–Pastur theorem; see e.g. Theorem 3.7 in . Also, from independence and bi-orthogonal invariance, it follows that for each $q=m-1,\ldots,1$, the matrices

are asymptotically free. Thus, the limiting singular value distribution of the product $\prod_{j=1}^{m}(\tau_{1}\mathbf{Y}^{\prime}_{j}+i\tau_{2}\mathbf{Y}^{\prime\prime}_{j})$ may be found by repeated application of Lemma A.2 in the appendix. Along these lines, many of the results in Section 8 may be extended to random matrices with a more general second moment structure as in (5.22).

Stieltjes Transforms of Spectral Limits of Shifted Matrices

In this section, we assume that $n=p$, i.e. $\mathbf{F}_{\mathbf{Y}}$ is a square matrix, and that the matrices $\mathbf{Y}^{(q)}$ have independent standard complex Gaussian entries, up to normalization.

where $\mathbf{J}(\alpha)$ is defined in Equation (6.1) below. More precisely, we will show that, under appropriate conditions, the mean eigenvalue distributions of the matrices $\mathbf{V}(\alpha)$ converge in moments to probability measures with compact support. Note that this implies the weak convergence of the mean eigenvalue distributions, and hence the weak convergence in probability of the eigenvalue distributions, by the variance estimate from Section A.1.

from the previous section. This matrix has spectral distribution $T(\alpha)=\frac{1}{2}\delta_{+|\alpha|}+\frac{1}{2}\delta_{-|\alpha|}$, where $\delta_{a}$ denotes the unit atom in the point $a$. We now calculate the $R$-transform of the distribution $T(\alpha)$. It is straightforward to check that for the distribution $T(\alpha)$, we have

(Recall that $M(z)$ and $G(z)$ have been introduced above Equation (5.1).) From (6.3) it follows that

Here we consider the principal branch of the square root. In order to obtain a function $R_{\alpha}(z)$ that is analytic at zero, we must take the plus sign. Therefore, (5.1) yields

Remark. Similarly, we find that the $S$-transform of the distribution $T(\alpha)$ is given by

We can now state the main result of this section.

Note that the measure $\mu_{\mathbf{V}}$ is symmetric with respect to the origin, and recall that in this case we choose the branch of $S$-transform $S_{\mathbf{V}}(z)$ as in (5.6).

By asymptotic freeness, the matrices $\mathbf{V}(\alpha)$ converge in moments to the probability measure $\mu_{\mathbf{V}(\alpha)}:=\mu_{\mathbf{V}}\boxplus T(\alpha)$; see the remark below Definition 5.5. Let $R_{\mathbf{V}(\alpha)}$ and $\widetilde{R}_{\mathbf{V}(\alpha)}$ denote the $R$-transform and $\widetilde{R}$-transform of $\mu_{\mathbf{V}(\alpha)}$, respectively. Then, by the additivity of the $\widetilde{R}$-transform (see Eq. (5.3)), we have

Using the relation (5.5) again, we finally get

Thus, Theorem 6.1 is proved completely. ∎

In the next section, we will consider the system (6.1) with $z=iy$, where $y\geq 0$ is any non-negative real number. The next results show that this is possible under appropriate conditions.

Let $\mu_{\mathbf{V}}$, $\mu_{\alpha}$ and $\mu_{\mathbf{V}(\alpha)}$ be defined as above.

For all $y>0$, $g(iy,\alpha)\in i[0,\tfrac{1}{2|\alpha|}]$.

The function $z\mapsto\widetilde{R}_{\alpha}(-g(z,\alpha))$ has an analytic continuation to an open neighborhood $U$ of the upper imaginary half-axis.

By abuse of notation, we still write $\widetilde{R}_{\alpha}(-g(z,\alpha))$ for the analytic continuation in part (iv), and we then define $w=w(z,\alpha)$ as in (6.1).

For $z=iy$ with $y>0$, we have the representation

where the branch of the square root is determined by the analytic continuation in part (iv). (Thus, the square root may be positive or negative!) In particular, $\widetilde{R}_{\alpha}(-g(iy,\alpha))\in$.

(i) follows from a straightforward calculation.

For the proof of (iii), note that (6.12) and (ii) imply

For the proof of (iv), recall that, for $z=iy$ with $y>0$ large enough,

This also establishes Equation (6.11). Since $h(iy)$ takes values in $$ by part (iii), the rest of part (v) follows immediately.

for all $y>0$, for it then follows by analytic continuation that the second equation in (6.1) holds for all $y>0$.

By the definition of $w(iy,\alpha)$, we have

Since $iyg(iy,\alpha)\in(-1,0)$ by part (i) and (6.13) and $\widetilde{R}_{\alpha}(-g(iy,\alpha))\in$ by part (v), it follows that

for $y>0$ large enough, and it remains to show (by continuity) that

for all $y>0$. Suppose by way of contradiction that $1+w(iy_{0},\alpha)g(iy_{0},\alpha)=0$ for some $y_{0}>0$. By (6.15), we may assume without loss of generality that $y_{0}>0$ is maximal with this property. Then $1+w(iy,\alpha)g(iy,\alpha)\in(0,1)$ for all $y>y_{0}$, and by analytic continuation, the second equation in (6.1) holds for all $y>y_{0}$. Letting $y\downarrow y_{0}$, we get

since $(-x)S_{\mathbf{V}}(-x)\to 0$ as $x\downarrow 0$. But this is a contradiction to (i). ∎

where $\widetilde{S}_{\mathbf{V}}(z):=zS_{\mathbf{V}}(z)$ for $z\in(-1,0)$ and $\widetilde{S}_{\mathbf{V}}(z)$ is defined by continuous extension for $z\in\{-1,0\}.$

We proceed by contradiction. Suppose that the limit $g(0,\alpha):=\lim_{y\downarrow 0}g(iy,\alpha)$ does not exist. Then, by Lemma 6.3 (iii) and continuity, the set of all accumulation points is a non-degenerate closed interval $I\subset i[0,\tfrac{1}{2|\alpha|}]$. But, as a consequence of Lemma 6.3 (vi), for each accumulation point $\widetilde{g}\in I\setminus\{0\}$, we have

where the square-root can be positive or negative. It is easy to see that this implies that

In view of our remark above Theorem 6.1, this means that $\mu_{\mathbf{V}}=T(\alpha)$, in contradiction to our assumption that $\mu_{\mathbf{V}}$ is not a two-point distribution. Thus, the limit $g(0,\alpha)$ exists, and (6.17) holds.

The existence of the limit $(wg)(0,\alpha):=\lim_{y\downarrow 0}(wg)(iy,\alpha)$ as well as the relations (6.4) are now simple consequences. It is worth noting here that the sign of the square root $\sqrt{1+4|\alpha|^{2}g(iy,\alpha)^{2}}$ can only change when $g(iy,\alpha)=\frac{i}{2|\alpha|}$, and hence must be constant for $y\approx 0$ when $g(0,\alpha)\neq\frac{i}{2|\alpha|}$. ∎

A similar argument shows that under the additional assumption that $g(iy,\alpha)$ is (jointly) continuous in $y$ and $\alpha$, we have

Note that Equation (6.17) has the “trivial” solution $g(0,\alpha)=0$ when the sign of the square-root is negative. The next result gives a sufficient condition for $g(0,\alpha)\neq 0$.

On the one hand, it is easy to see that there exists a constant $C>0$ such that

for all sufficiently small $r>0$. On the other hand, our assumption implies that

Thus we find that the square-root must be negative for all sufficiently small $y>0$. Using Taylor expansion, it follows that

Recalling that $1+(wg)(iy,\alpha)$ takes values in $$, we may conclude that for all sufficiently small$y>0$,

By (6.1) and (6.18), it follows that for all sufficiently small $y>0$,

For $|\alpha|<\frac{1}{\sqrt{2}C}$, this is a contradiction. Consequently, our assumption that $g(0,\alpha)=0$ is wrong in this case, and Lemma 6.6 is proved. ∎

Density of Limiting Spectral Distribution

In this section, we compute the density of the limit distribution of the empirical spectral distributions of the matrices $\mathbf{F}_{\mathbf{Y}}$. Here we assume that $n=p$, i.e. $\mathbf{F}_{\mathbf{Y}}$ is a square matrix, and that the matrices $\mathbf{Y}^{(q)}$ have independent standard complex Gaussian entries, up to normalization.

To study the limiting distribution of the eigenvalue distributions of the matrices $\mathbf{F}_{\mathbf{Y}}$, we use the method of hermitization which goes back to Girko . This method may be summarized as follows:

See e.g. Lemma 4.3 in Bordenave and Chafaï . Let us mention here that Conditions $(C0)$, $(C1)$ and $(C2)$ together with the assumption of weak convergence in probability imply that the function $\log|\,\cdot\,|$ is uniformly integrable in probability for the measures $\nu_{n}(\,\cdot\,,\alpha)$ and that the integrals in (7.1) are finite; see also Lemma A.9 in Appendix A.4.

We now describe the density $f$ of the limiting spectral distribution $\mu_{\mathbf{F}}$ in terms of the $S$-transform of the measure $\mu_{\mathbf{V}}$. In doing so, we will not use any special properties of random matrices, but only the probability measures $\mu_{\mathbf{V}}$ and $\mu_{\mathbf{F}}$ and their properties stated below.

We shall additionally make the following assumptions:

where the square-root is the same as in (6.11). Moreover, the function $g(iy,\alpha)$ admits a continuous extension $g(0,\alpha)$ as $y\downarrow 0$.

The following lemma shows that Assumptions 7.2 and 7.3 are satisfied if the probability measure $\mu_{\mathbf{V}}$ has compact support or, more generally, sufficiently small tails. Since the proof is rather technical, it is deferred to Appendix A.4.

Assumptions 7.2 and 7.3 hold for probability measures $\mu_{\mathbf{V}}$ such that $\mu_{\mathbf{V}}([-x,+x]^{c})=\mathcal{O}(x^{-\eta})$ $(x\to\infty)$ for some $\eta>0$.

The logarithmic transform of the measures $\nu(\,\cdot\,\alpha)$ is defined by

Note that this is exactly the integral on the right-hand side in (7.3). Similarly as above, we regard the function $\Phi$ as a function of the real parameters $u$ and $v$.

where the function $g(0,\alpha)$ and the sign of the square-root are the same as in (6.17).

Clearly, by Assumption 7.3, the expression in the second line can be made arbitrarily small by choosing $C$ sufficiently large. Also, note that $\varkappa(C,\alpha)\to 0$ as $C\to\infty$. Thus, to complete the proof of the lemma, it remains to show that for any $C>0$,

Let us mention here that the sign of the first square-root is positive for $C$ large enough, whereas the sign of the second square-root may be positive or negative, as in Lemma 6.4.

To prove (7.7), note that by Assumption 7.2, we have

for any $0<c<C$. It therefore follows that

Thus, setting $c:=|h|$ and letting $h\to 0$, we obtain

where we have used the fact that the functions $\varkappa(y,\alpha)$ and $\sqrt{1-4|\alpha|^{2}\varkappa^{2}(y,\alpha)}$ are bounded and continuous near the point $(0,\alpha)$. This completes the proof of (7.7). ∎

Suppose that $\mu_{\mathbf{V}}$ and $\mu_{\mathbf{F}}$ are as above and that Assumptions 7.2 and 7.3 hold. For $x>0$ and $\alpha\neq 0$, introduce the functions

where $w(z,\alpha)$ is defined as in Theorem 6.1, and their limits

Then the functions $\varkappa(\,\cdot\,,\alpha)$ and $\psi(\,\cdot\,,\alpha)$ are real-valued with values in $[0,\tfrac{1}{2\alpha}]$ and $$, respectively. Furthermore, set$\xi_{\mathbf{V}}(x):=i(-x)S_{\mathbf{V}}(-x)$$(x\in[0;1])$. Then$\xi_{\mathbf{V}}\geq 0$, and we have the relation

Alternatively, and more conveniently for applications, we may rewrite Equation (7.8) in the form of two equations:

One nuisance is that the solution to (7.6) is not unique. Indeed, the pair $(\varkappa,\psi)\equiv(0,1)$ is always a solution. However, this trivial solution can be excluded using Lemma 6.6. In typical applications, we will proceed as follows: For given $\xi_{\mathbf{V}}$, solve the system (7.6). Using Lemma 6.6, argue that the solution is unique. This is possible at least in some cases, and notably in all our applications. Then check that the unique solution is continuously differentiable (except on a finite number of rings) and compute $f$ using (7.10). Finally, check that $f$ is indeed a probability density.

Note that the measure $\nu(\cdot,\alpha)$ with corresponding Stieltjes transform $g(z,\alpha)$ is symmetric to the origin. Thus, by Lemma 6.3 (iii) and (vi), we have $\varkappa(x,\alpha)\in[0,\tfrac{1}{2\alpha}]$ and $\psi(x,\alpha)\in$ for all $x\geq 0$. Also, the limits $\varkappa(0,\alpha)$ and $\psi(0,\alpha)$ exist by Lemma 6.4 and the subsequent remark. Furthermore, by our conventions concerning the $S$-transform, we have $\xi_{\mathbf{V}}(x)\geq 0$ for all $x\in(0,1)$.

We now rewrite the Equations (6.1) in terms of the real-valued functions $\varkappa$, $\psi$ and $\xi_{\mathbf{V}}$. Using (6.1) with $z=ix$, we have

where the sign of the square-root is determined as in (6.11). Letting $x\downarrow 0$, we get

where the sign of the square-root is determined by continuous extension. Taking squares in the first equation and rearranging terms, we deduce that

Eliminating $\varkappa$ from these equations leads to the equivalent equation (7.8).

Suppose additionally that there exists a finite set $A$ such that for $\alpha\not\in A$, the function $\psi(\alpha)$ is continuously differentiable at $\alpha$. Let $\alpha\not\in A$. By Lemma 7.5 and Equation (7), we have

Since $\psi$ is continuously differentiable w.r.t. $u$, it follows that

Note that all functions depend on $|\alpha|$ only, and are therefore symmetric with respect to $u$ and $v$. Thus we also have

Summing (7.15) and (7.16), we get, for $|\alpha|\not\in A$,

Moreover, it follows from the preceding discussion that on the open set where $|\alpha|\not\in A$, $\Phi(\alpha)$ is twice continuously differentiable.

In view of relation (7.3), this means that the log-potential $U_{\mathbf{F}}(\alpha)$ is twice continuously differentiable on the open set where $|\alpha|\not\in A$. It therefore follows by a well known result from potential theory (see e.g. , Theorem II.1.3) that the restriction of $\mu_{\mathbf{F}}$ to this set is absolutely continuous with Lebesgue density

This completes the proof of Theorem 7.6. ∎

Let us mention that in Chapter II.1 in , it is assumed that the measure $\mu$ under consideration is finite and of compact support. However, a closer inspection of the proof shows that the latter assumption may be relaxed; it is sufficient to assume that the function $z\mapsto\log^{+}|z|$ is integrable w.r.t. $\mu$.

Applications

We consider applications of Theorems 3.2 and 4.4. These applications show that our main results allow old and new results on products of independent random matrices to be derived in a unified way. In doing so, we shall always assume that either all random variables $X_{jk}^{(q)}$ are real with

or all random variables $X_{jk}^{(q)}$ are complex with

Clearly, under these assumptions, the corresponding random variables $Y_{jk}^{(q)}$ as in (2) have standard real or complex Gaussian distributions, respectively, and we may use the results from the preceding sections. Let us note here that although we have stated these results only for the complex case, there exist analogous results for the real case, as mentioned at the end of Section 5. Furthermore, let us emphasize that the assumptions (8.1) and (8.2) are not needed to establish universality, but only to identify the limiting distributions. Finally, let us mention that the assumptions (8.1) and (8.2) can be relaxed a bit; see the Remark at the end of Section 5.

In this section we consider some applications of Theorem 3.2. We start from the simplest case of Marchenko–Pastur law.

Assume that the random variables $X_{jk}$, $j=1,\ldots,n$, $k=1,\ldots,p$ satisfy the Lindeberg condition (3.1). Then

where $G^{\prime}(x)=\frac{\sqrt{(x-a)(b-x)}}{2\pi xy}$ with $a=(1-\sqrt{y})^{2}$, $b=(1+\sqrt{y})^{2}$.

The probability distribution given by the distribution function $G(x)$ is called the Marchenko–Pastur distribution with parameter $y$.

For simplicity we shall consider only the case that the r.v.’s $X_{jk}$ are real. Let $\mathbf{Y}$ be a Gaussian matrix as in Section 1. We prove only the universality of the singular value distribution of the matrix $\mathbf{X}$, and then suppose that the limiting distribution of the singular values of the Gaussian matrix $\mathbf{Y}$ is known.

To apply Theorem 3.2, we check conditions (3.2), (3.20), (3.22), and (3.24). First we note that in our case

Starting from (8.5), it is straightforward to check that condition (3.20) holds with constant $A_{0}=2v^{-2}$, condition (3.22) holds with constant $A_{1}=8v^{-3}$ and condition (3.24) holds with constant $A_{2}=48v^{-4}$.

where $a=(1-\sqrt{y})^{2}$ and $b=(1+\sqrt{y})^{2}$. Thus Theorem 8.1 is proved. ∎

The $S$-transform of the Marchenko–Pastur distribution with parameter $y\in(0,1]$, with density defined in (8.6), is given by

Let $g_{y}(z)$ denote the Stieltjes transform of the Marchenko–Pastur distribution. It is well-known that

See for instance , Equations (3.1) and (3.9). Using this equation and the formal identity

Solving this equation with respect to $z$, we obtain

Let $X$ be a r.v. with Marchenko-Pastur distribution with parameter $y=1$, and let $F_{t}$ denote the distribution of $(X+t)^{-1}$, $t\geq 0$. Then $F_{t}\to F$ in Kolmogorov distance as $t\to 0$, and the $S$-transform of the limit $X^{-1}$ is given by

The first part follows from the pointwise convergence of the corresponding densities, which are easily calculated using (8.6). For the second part, we provide a formal proof. Recall that $S_{X}(z)=\frac{1}{z+1}$. The corresponding Stieltjes transform is $g_{X}(z)=\frac{1}{2}(-1+\sqrt{\frac{z-4}{z}})$. Furthermore, we note that formally $M_{X^{-1}}(z)=zg_{X}(z)$, where $M_{X^{-1}}(z)$ denotes the generic moment generating function of the distribution of $X^{-1}$. This implies

1.2 Product of Independent Rectangular Matrices

Let $m\geq 1$ be fixed. Let $n_{0},\ldots,n_{m}$ denote integers depending on $n\geq 1$ such that $n_{0}=n$ and

Assume that the random variables $X_{jk}^{(q)}$, for $q=1,\ldots,m$ and $j=1,\ldots,n_{q-1}$; $k=1,\ldots,n_{q}$, satisfy the Lindeberg condition (3.1). Then

where the Stieltjes transform $s(z)=\int_{-\infty}^{\infty}\frac{1}{x-z}dG^{(m)}(x)$ of the distribution function $G^{(m)}(x)$ is determined by the equation

For simplicity we shall assume that all r.v.’s $X^{(q)}_{jk}$ are real. Let $\mathbf{Y}^{(1)},\ldots,\mathbf{Y}^{(q)}$ be Gaussian matrices as in Section 1.

Let $\mathbf{Z}^{(1)},\ldots,\mathbf{Z}^{(q)}$ be defined as in (3.17). Furthermore, introduce the matrices

for $j=1,\ldots,n_{q-1};k=1,\ldots,n_{q}$. From here it follows that

Consider for instance the first term in the right hand side of (8.9). We have

Note that for each $q=1,\ldots,m$, the vector $\mathbf{V}_{1,q-1}\mathbf{e}_{j}^{(q)}$ is independent of the matrix $\mathbf{Z}^{(q)}$ due to the block structure of the matrices $\mathbf{H}^{(q)}$. From here it follows that

and by Lemma 5.1 in the Appendix of , we have

$q=1,\ldots,m$. Combining these estimates, it follows that

Furthermore, it is straightforward to check that

Using equalities (8.12), (8.8), (8.10) and Lemma 5.1 in the Appendix of , we get

Using equalities (8.13), (8.8), (8.10) and Lemma 5.2 in the Appendix of , it is straightforward to prove that

Inequalities (8.11), (8.12), (8.14) imply that conditions (3.20), (3.22), (3.24) hold. Thus, from Theorem 3.2, it follows that the limit distribution of the singular values of the matrices $\mathbf{F}_{\mathbf{X}}$ is the same as the limit distribution of the singular values of the matrices $\mathbf{F}_{\mathbf{Y}}$. For the Gaussian case we may prove that the random matrices $(\prod_{q=1}^{l-1}\mathbf{Y}^{(q)})^{*}(\prod_{q=1}^{l-1}\mathbf{Y}^{(q)})$ and $\mathbf{Y}^{(l)}{\mathbf{Y}^{(l)}}^{*}$ are asymptotically free for any $l=1,\ldots,m$. For details see , Lemma 4.1. From here and Lemma A.2 it follows that the $S$-transform of the distribution function $G^{(m)}(x)$ is given by

For details, see Equations (4.9) and (4.13) in . Thus Theorem 8.5 is proved. ∎

Assume that the conditions of Theorem 8.5 hold and $y_{1}=\cdots=y_{m}=1$. Then

where the Stieltjes transform of $G(x)$ is determined by the equation

1.3 Powers of Random Matrices

Assume that the random variables $X_{jk}$ satisfy the Lindeberg condition (3.1). Then

where $G^{(m)}(x)$ is defined by its Stieltjes transform $s(z)$, which satisfies the equation

The numbers $M_{k}$ appearing in (8.17) are called Fuss–Catalan numbers.

Again, for simplicity we shall consider only the case that the r.v.’s $X_{jk}$ are real. Let $\mathbf{Y}$ be a Gaussian matrix as in Section 1.

where $\mathbf{H}=\begin{bmatrix}&\mathbf{Z}&\mathbf{O}&\\ &\mathbf{O}&\mathbf{Z}^{*}&\end{bmatrix}$ and $\mathbf{J}=\mathbf{J}(-1)$. Clearly,

Consider for instance the first sum on the right-hand side. Applying Hölder’s inequality, we get

where $\mathbf{H}^{(j,k)}$ is obtained from $\mathbf{H}$ by replacing the entry $Z_{jk}$ with zero. Note that, for $q\geq 2$,

By the independence of the matrices $\mathbf{H}^{(j,k)}$ and the random variables $Z_{jk}$, we have

for some absolute positive constants $C_{1},C_{2}$. Similarly we have

for some positive constant $C>0$. (Let us note here that the moment conditions here are a bit different from those in , but using (3.9) – (3.12), it is easy to see that the conclusion still holds.) Furthermore,

Thus, we may rewrite the equality (8.19) in the form

All summands on the r.h.s of (8.1.3) may be bounded similarly. For instance,

where the sum is taken over all $t,u,v,w$ from the set $\{j,k,j+n,k+n\}$. Applying Hölder’s inequality, the representation (8.18) and Lemmas 5.2 and 5.4 in , we get