Edgeworth-type expansion in the entropic free CLT

TL;DR

This paper develops an Edgeworth-type expansion for densities in the free central limit theorem and applies it to the entropic free CLT under a fourth-moment condition, enhancing understanding of convergence rates.

Contribution

It introduces a new density expansion in the free CLT and extends it to the entropic case with moment assumptions, advancing free probability theory.

Findings

Derived an Edgeworth-type expansion for free CLT densities

Applied the expansion to the entropic free CLT under fourth-moment conditions

Provided insights into convergence behavior in free probability

Abstract

We prove an expansion for densities in the free CLT and apply this result to an expansion in the entropic free central limit theorem assuming a moment condition of order four for the free summands.

Equations

$$Y_n:=\frac{X_1+X_2+\cdots +X_n}{\sqrt{n}}$$

$$p_n(x)\le c(\mu ),x\in \mathbb{R}.$$

$$a_n:=\frac{m_3(\mu )}{\sqrt{n}},b_n:=\frac{m_4(\mu )-m_3^2(\mu )-1}{n},d_n:=\frac{m_4(\mu )-m_3^2(\mu )}{n},n\in \mathbb{N}.$$

$$I_{n}:=\Big{\{}x\in\mathbb{R}:|x-a_{n}|\leq\frac{2}{e_{n}}-\frac{\varepsilon_{n1}}{n}\Big{\}},\qquad I_{n}^{*}:=\Big{\{}x\in\mathbb{R}:|x-a_{n}|\leq\frac{2}{e_{n}}-\sqrt{\frac{\varepsilon_{n1}}{n}}\Big{\}}.$$

$$v_{n}(x)=\Big{(}1+\frac{1}{2}d_{n}-a_{n}^{2}-\frac{1}{n}-a_{n}x-\Big{(}b_{n}-a_{n}^{2}-\frac{1}{n}\Big{)}x^{2}\Big{)}p_{w}(e_{n}x),\quad x\in\mathbb{R}.$$

$$p_n(x+a_n)=v_n(x)+{\rho }_{n1}(x)+{\rho }_{n2}(x),x\in I_n^{\ast }-a_n,$$

$$|{\rho }_{n1}(x)|\le \frac{{\epsilon }_{n1}}{n}\frac{c(\mu )}{(4-(e_{n}x{)}^2{)}^{3/2}},$$

$$|{\rho }_{n1}(x)|\le \sqrt{\frac{{\epsilon }_{n1}}{n}}\frac{c(\mu )}{(4-(e_{n}x{)}^2{)}^{1/2}}.$$

$$0\le {\rho }_{n2}(x)\le c(\mu )\text{ and}{\int }_{I_n-a_n}{\rho }_{n2}(x)dx=o(1/n^2).$$

$${\int }_{\mathbb{R}\setminus I_n}{p}_n(x)dx\le \frac{{\epsilon }_{n2}}{n}.$$

$$\int_{\mathbb{R}}|p_{n}(x)-p_{w}(x)|\,dx=\frac{2|m_{3}(\mu)|}{\pi\sqrt{n}}+c(\mu)\theta\Big{(}\Big{(}\frac{\varepsilon_{n1}}{n}\Big{)}^{3/4}+\frac{1}{n}\Big{)}.$$

$$D(X)=D(X||Z)=h(Z)-h(X),$$

$$\chi (\nu )=\int {\int }_{\mathbb{R}\times \mathbb{R}}\log |x-y|\nu (dx)\nu (dy)+\frac{3}{4}+\frac{1}{2}\log 2\pi ,$$

$$D(\nu ||{\mu }_w)=\chi ({\mu }_w)-\chi (\nu )$$

$$\displaystyle D(\mu_{n}||\mu_{w})=\frac{m_{3}^{2}(\mu)}{6n}+\theta\Big{(}c(\mu,q)\Big{(}\frac{\varepsilon_{n1}}{n}\Big{)}^{1+\frac{1}{2q}}+c(\mu)\frac{\varepsilon_{n2}}{n}\Big{)},$$

$$I(X||Z)=I(X)-I(Z)$$

$$\Phi (\nu )=\frac{4{\pi }^2}{3}{\int }_{\mathbb{R}}p(x{)}^{3}dx.$$

$$\Phi (\nu ||{\mu }_w)=\Phi (\nu )-\Phi ({\mu }_w)\ge 0$$

$$\Phi ({\mu }_n||{\mu }_w)={\int }_{\mathbb{R}}{p}_n(x{)}^{3}dx-\Phi ({\mu }_w)=\frac{m_3^2(\mu )}{n}+c(\mu )\theta \frac{{\epsilon }_{n1}+{\epsilon }_{n2}}{n}.$$

$$f(z)=a+bz+\int\limits_{\mathbb{R}}\frac{1+uz}{u-z}\,\tau(du)=a+bz+\int\limits_{\mathbb{R}}\Big{(}\frac{1}{u-z}-\frac{u}{1+u^{2}}\Big{)}(1+u^{2})\,\tau(du),\quad z\in\mathbb{C}^{+},$$

$$f(z)=\underset{\mathbb{R}}{\int }\frac{\sigma (du)}{u-z},z\in {\mathbb{C}}^{+},$$

$$G_{\mu }(z)={\int }_{\mathbb{R}}\frac{\mu (du)}{z-u},z\in {\mathbb{C}}^{+}.$$

$${\mu }_y(dx)=-\frac{1}{\pi }\Im G_{\mu }(x+iy)dx,x\in \mathbb{R},y>0,$$

$$D_{\mu }^{\prime }(x)=-\Im G_{\mu }(x)/\pi .$$

$$F_{\mu }(z)=\frac{1}{G_{\mu }(z)}.$$

$$m_k=m_k(\mu ):=\underset{\mathbb{R}}{\int }{u}^k\mu (du)<\infty ,k=0,1,\dots ,2n,n\ge 1.$$

$$\lim_{z\to\infty}z^{2n+1}\Big{(}G_{\mu}(z)-\frac{1}{z}-\frac{m_{1}}{z^{2}}-\dots-\frac{m_{2n-1}}{z^{2n}}\Big{)}=m_{2n}$$

$$F_{\mu }(z)=z+{\int }_{\mathbb{R}}\frac{\tau (du)}{u-z},z\in {\mathbb{C}}^{+},$$

$$m_k(\mu )=\sum _{l=1}^{[k/2]}\sum _{s_1+\cdots +s_l=k-2,s_j\ge 0}{m}_{s_1}(\tau )\dots m_{s_l}(\tau ),k=2,\dots ,2n.$$

$$z=Z_1(z)+Z_2(z)-F_{{\mu }_1}(Z_1(z))\text{and}{F}_{{\mu }_1}(Z_1(z))=F_{{\mu }_2}(Z_2(z)).$$

Full text

1. Introduction

Free convolutions were introduced by D. Voiculescu [32], [33] and have been studied intensively in context of non commutative probability. The key concept here is the notion of freeness, which can be interpreted as a kind of independence for non commutative random variables. As in classical probability theory where the concept of independence gives rise to the classical convolution, the concept of freeness leads to a binary operation on the probability measures, the free convolution. Many classical results in the theory of addition of independent random variables have their counterparts in Free Probability, such as the Law of Large Numbers, the Central Limit Theorem, the Lévy-Khintchine formula and others. We refer to Voiculescu, Dykema and Nica [34], Hiai and Petz [20], and Nica and Speicher [26] for an introduction to these topics.

In this paper we obtain an analogue of Esseen’s expansion for a density of normalized sums of free identically distributed random variables under a fourth moment assumption on the free summands. Using this expansion we establish the rate of convergence of the free entropy of normalized sums of free identically distributed random variables.

The paper is organized as follows. In Section 2 we formulate and discuss the main results of the paper. In Section 3 and 4 we state auxiliary results. In Section 5 we discuss the passage to probability measures with bounded supports. In Section 6 we obtain a local asymptotic expansion for a density in the CLT for free identically distributed random variables. In Section 7 we study the behaviour of subordination functions in the free CLT for truncated free summands. In Section 8 we discuss the closeness of subordination functions in the free CLT for bounded and unbounded free random variables. In Section 9 we investigate the rate of convergence for densities in the free CLT in $L_{1}(-\infty,+\infty)$ and Section 10 is devoted to study the rate of convergence for the free entropy of normalized sums of free identically distributed random variables. In Section 11 we derive rates of convergence for the free Fisher information of normalized sums of free identically distributed random variables.

2. Results

Denote by $\mathcal{M}$ the family of all Borel probability measures defined on the real line $\mathbb{R}$. Let $\mu\boxplus\nu$ be the free (additive) convolution of $\mu$ and $\nu$ introduced by Voiculescu [32] for compactly supported measures. Free convolution was extended by Maassen [24] to measures with finite variance and by Bercovici and Voiculescu [8] to the class $\mathcal{M}$. Thus, $\mu\boxplus\nu=\mathcal{L}(X+Y)$, where $X$ and $Y$ are free random variables such that $\mu=\mathcal{L}(X)$ and $\nu=\mathcal{L}(Y)$.

Henceforth $X,X_{1},X_{2},\dots$ stands for a sequence of identically distributed random variables with distribution $\mu=\mathcal{L}(X)$. Define $m_{k}(\mu):=\int_{\mathbb{R}}u^{k}\,\mu(du)$, where $k=0,1,\dots$.

The classical CLT says that if $X_{1},X_{2},\dots$ are independent and identically distributed random variables with a probability distribution $\mu$ such that $m_{1}(\mu)=0$ and $m_{2}(\mu)=1$, then the distribution function $F_{n}(x)$ of

[TABLE]

tends to the standard Gaussian law $\Phi(x)$ as $n\to\infty$ uniformly in $x$.

A free analogue of this classical result was proved by Voiculescu [31] for bounded free random variables and later generalized by Maassen [24] to unbounded random variables. Other generalizations can be found in [9], [10], [16], [21]–[23], [27], [37], [38].

For $t>0$, the centered semicircle distribution of variance $t$ is the probability measure with density $p_{w_{t}}(x):=\frac{1}{2\pi t}\sqrt{(4t-x^{2})_{+}},\,x\in\mathbb{R}$, where $a_{+}:=\max\{a,0\}$ for $a\in\mathbb{R}$. Denote by $\mu_{w_{t}}$ the probability measure with the distribution function $w_{t}(x)$. In the sequel we use the notations $w_{1}(x)=w(x)$.

When the assumption of independence is replaced by the freeness of the non commutative random variables $X_{1},X_{2},\dots,X_{n}$, the limit distribution function of (2.1) is the semicircle law $w(x)$. We denote as well by $\mu_{n}$ the probability measure with the distribution function $F_{n}(x)$.

It was proved in [5] that if the distribution $\mu$ of $X$ is not a Dirac measure, then in the free case $F_{n}(x)$ is Lebesgue absolutely continuous when $n\geq n_{1}=n_{1}(\mu)$ is sufficiently large. Denote by $p_{n}(x)$ the density of $F_{n}(x)$.

In the sequel we denote by $c(\mu),c_{1}(\mu),c_{2}(\mu),\dots$ positive constants depending on $\mu$ only. By $c(\mu)$ we denote generic constants in different (or even in the same) formula. The symbols $c_{1}(\mu),c_{2}(\mu),\dots$ will denote explicit constants. By $\{\varepsilon_{nk}\}$ denote positive numbers such that $\varepsilon_{nk}\to 0$ as $n\to\infty$.

Wang [38] proved that under the condition $m_{2}(\mu)<\infty$ the density $p_{n}(x)$ of $F_{n}(x)$ is continuous for sufficiently large $n$ and

[TABLE]

Assume that $m_{4}(\mu)<\infty,m_{1}(\mu)=0,m_{2}(\mu)=1$ and denote

[TABLE]

Furthermore, let $e_{n}:=(1-b_{n})/\sqrt{1-d_{n}}$ and let $I_{n}$ and $I_{n}^{*}$ denote intervals of the form

[TABLE]

In the sequel we denote by $\theta$ a real-valued quantity such that $|\theta|\leq 1$.

We have derived an asymptotic expansion of $p_{n}(x)$ for bounded free random variables $X_{1},X_{2},\dots$ in the paper [17]. Improving the methods of this paper and [18] we obtain an asymptotic expansion of $p_{n}(x)$ for the case $m_{4}(\mu)<\infty$. Denote by $v_{n}(x)$ the function

[TABLE]

Theorem 2.1**.**

Let $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$. Then there exist sequences $\{\varepsilon_{n1}\}$ and $\{\varepsilon_{n2}\}$ such that

[TABLE]

where, for $x\in I_{n}^{*}-a_{n}$,

[TABLE]

and for $x\in(I_{n}-a_{n})\setminus(I_{n}^{*}-a_{n})$,

[TABLE]

In $(\ref{asden})$ $\rho_{n2}(x)$ is a continuous function such that

[TABLE]

Moreover,

[TABLE]

Corollary 2.2**.**

Let $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$, then

[TABLE]

In [17] we proved analogous results for bounded free random variables and in [18] assuming a finite moment of order eight.

Recall that, if the random variable $X$ has density $f$, then the classical entropy of a distribution of $X$ is defined as $h(X)=-\int_{\mathbb{R}}f(x)\log f(x)\,dx$, provided the positive part of the integral is finite. Thus we have $h(X)\in[-\infty,\infty)$. A much stronger statement than the classical CLT – the entropic central limit theorem – indicates that, if for some $n_{0}$, or equivalently, for all $n\geq n_{0}$, $Y_{n}$ from (2.1) have absolutely continuous distributions with finite entropies $h(Y_{n})$, then there is convergence of the entropies, $h(Y_{n})\to h(Y)$, as $n\to\infty$, where $Y$ is a standard Gaussian random variable. This theorem is due to Barron [3]. Artstein, Bally, Barthez, and Naor [2] have solved an old question raised by Shannon about the monotonicity of entropy under convolution. The relative entropy

[TABLE]

where the normal random variable $Z$ have the same mean and the same variance as $X$, is nonnegative and serves as kind of a distance to the class of normal laws. Thus, the entropic central limit theorem may be reformulated as $D(Y_{n})\downarrow 0$, as long as $D(Y_{n_{0}})<+\infty$ for some $n_{0}$.

Recently Bobkov, Chistyakov and Götze [12] found the rate of convergence to zero of $D(Y_{n})$ and for the random variables $X$ with ${\mathbb{E}}|X|^{k}<\infty,\,k\geq 4$, have obtained an Edgeworth-type expansion of $D(Y_{n})$ as $n\to\infty$.

Let $\nu$ be a probability measure on $\mathbb{R}$. We assume below that $m_{1}(\nu)=0$ and $m_{2}(\nu)=1$. The quantity

[TABLE]

called free entropy, was introduced by Voiculescu in [35]. Free entropy $\chi$ behaves like the classical entropy $h$. In particular, the free entropy is maximized by the standard semicircular law $w$ with the value $\chi(w)=\frac{1}{2}\log 2\pi e$ among all probability measures with variance one [20], [36]. Shlyakhtenko [29] has proved that $\chi(\mu_{n})$ decreases monotonically, i.e., the Shannon hypothesis holds in the free case as well.

Wang [38] has proved a free analogue of Barron’s result: the free entropy $\chi(\mu_{n})$ converges to the semicircular entropy. As in the classical case a relative free entropy

[TABLE]

is nonnegative and serves as kind of a distance to the class of semicircular laws.

We derive an optimal rate of convergence in the free CLT for free random variables with a finite moment of order four. In previous results [17] we showed an analogous result for bounded free random variables and in [18] for free random variables with a finite moment of order eight.

Corollary 2.3**.**

Let $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$. Then, for every fixed $1<q\leq 1.01$,

[TABLE]

where $c(\mu,q)>0$ is a constant depended on $\mu$ and $q$ only.

Hence the remainder term in (2.12) is of order $o(n^{-1})$ provided that $m_{4}(\mu)<\infty$. In Sections 5 and 8 we explicitly describe the sequences $\{\varepsilon_{n1}\}$ and $\{\varepsilon_{n2}\}$. If we assume that $m_{6}(\mu)<\infty$, then it follows from Remarks 8.13 and 8.14 (see the end of Section 8) that the remainder term in (2.12) is of order $O(n^{-3/2})$.

Given a random variable $X$ with an absolutely continuous density $f$, the Fisher information of $X$ is defined by $I(X)=\int_{-\infty}^{+\infty}\frac{f^{\prime}(x)^{2}}{f(x)}\,dx$, where $f^{\prime}$ denotes the Radon-Nikodym derivative of $f$. In all other cases, let $I(X)=+\infty$. With the first two moments of $X$ being fixed, $I(X)$ is minimized for the normal random variable $Z$ with the same mean and the same variance as $X$, i.e. $I(X)\geq I(Z)$ (which is a variant of Cramér-Rao’s inequality).

Baron and Johnson have proved in [4] that $I(Y_{n})\to I(Z)$, as $n\to\infty$, if and only if $I(Y_{n_{0}})<\infty$. In classical probability and statistics the relative Fisher information

[TABLE]

is used as a strong measure of the probability distribution of $X$ being near to the Gaussian distribution. The result of Baron and Johnson is equivalent to the fact that $I(Y_{n}||Z)\to 0$ as $n\to\infty$, if and only if $I(Y_{n_{0}}||Z)<\infty$.

Bobkov, Chistyakov and Götze [13] found the rate of convergence to zero of $I(Y_{n}||Z)$ and for the random variables $X$ with ${\mathbb{E}}|X|^{k}<\infty,\,k\geq 4$, have obtained an Edgeworth-type expansion of $I(Y_{n}||Z)$ as $n\to\infty$.

Suppose that the measure $\nu$ has a density $p$ in $L^{3}(\mathbb{R})$. Then, following Voiculescu [36], the free Fisher information is

[TABLE]

It is well-known that $\Phi(\mu_{w})=1$. The free Fisher information has many properties analogous to those of classical Fisher information. These include the free analog of the Cramér-Rao inequality.

Assume now that $m_{1}(\nu)=0$ and $m_{2}(\nu)=1$. Consider the free relative Fisher information

[TABLE]

as a strong measure of closeness of $\nu$ to Wigner’s semicircle law. Here we obtain an Edgeworth-type expansion for free random variables with a finite moment of order four.

Corollary 2.4**.**

Let $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$. Then

[TABLE]

As in the formula (2.12) the remainder term here is of order $o(n^{-1})$ if $m_{4}(\mu)<\infty$ and of order $O(n^{-3/2})$ provided that $m_{6}(\mu)<\infty$.

In contrast to the classical case (see [12] and [13]) we expect that the asymptotic expansion in (2.12) and (2.13) holds with an error of order $n^{-1}$ only.

3. Auxiliary results

We need results about some classes of analytic functions (see [1], Section 3.

The class $\mathcal{N}$ (Nevanlinna, R.) is the class of analytic functions $f(z):\mathbb{C}^{+}\to\{z:\,\Im z\geq 0\}$. For such functions there is an integral representation

[TABLE]

where $b\geq 0$, $a\in\mathbb{R}$, and $\tau$ is a non-negative finite measure. Moreover, $a=\Re f(i)$ and $\tau(\mathbb{R})=\Im f(i)-b$. From this formula it follows that $f(z)=(b+o(1))z$ for $z\in\mathbb{C}^{+}$ such that $|\Re z|/\Im z$ stays bounded as $|z|$ tends to infinity (in other words $z\to\infty$ non tangentially to $\mathbb{R}$). Hence if $b\neq 0$, then $f$ has a right inverse $f^{(-1)}$ defined on the region $\Gamma_{\alpha,\beta}:=\{z\in\mathbb{C}^{+}:|\Re z|<\alpha\Im z,\,\Im z>\beta\}$ for any $\alpha>0$ and some positive $\beta=\beta(f,\alpha)$.

A function $f\in\mathcal{N}$ admits the representation

[TABLE]

where $\sigma$ is a finite non-negative measure, if and only if $\sup_{y\geq 1}|yf(iy)|<\infty$. Moreover $\sigma(\mathbb{R})=-\lim_{y\to+\infty}iyf(iy)$.

For $\mu\in\mathcal{M}$, consider its Cauchy transform $G_{\mu}(z)$

[TABLE]

The measure $\mu$ can be recovered from $G_{\mu}(z)$ as the weak limit of the measures

[TABLE]

as $y\downarrow 0$. If the function $\Im G_{\mu}(z)$ is continuous at $x\in\mathbb{R}$, then the probability distribution function $D_{\mu}(t)=\mu((-\infty,t))$ is differentiable at $x$ and its derivative is given by

[TABLE]

This inversion formula allows to extract the density function of the measure $\mu$ from its Cauchy transform.

Following Maassen [24] and Bercovici and Voiculescu [8], we shall consider in the following the  reciprocal Cauchy transform

[TABLE]

The corresponding class of reciprocal Cauchy transforms of all $\mu\in\mathcal{M}$ will be denoted by $\mathcal{F}$. This class coincides with the subclass of Nevanlinna functions $f$ for which $f(z)/z\to 1$ as $z\to\infty$ non tangentially to $\mathbb{R}$.

The following lemma is well-known, see [1], Th. 3.2.1, p. 95.

Lemma 3.1**.**

Let $\mu$ be a probability measure such that

[TABLE]

Then the following relation holds

[TABLE]

uniformly in the angle $\delta\leq\arg z\leq\pi-\delta$, where $0<\delta<\pi/2$.

Conversely, if for some function $G(z)\in\mathcal{N}$ the relation $(\ref{3.4abl2})$ holds with real numbers $m_{k}$ for $z=iy,y\to\infty$, then $G(z)$ admits the representation $(\ref{3.5a})$, where $\mu$ is a probability measure with moments $(\ref{3.4abl1})$.

As shown before, $F_{\mu}(z)$ admits the representation (3.1) with $b=1$. From Lemma 3.1 the following proposition is immediate.

Proposition 3.2**.**

In order that a probability measure $\mu$ satisfies the assumption $(\ref{3.4abl1})$, where $m_{1}(\mu)=0$, it is necessary and sufficient that

[TABLE]

where $\tau$ is a nonnegative measure such that $m_{2n-2}(\tau)<\infty$. Moreover

[TABLE]

Voiculescu [35] showed for compactly supported probability measures that there exist unique functions $Z_{1},Z_{2}\in\mathcal{F}$ such that $G_{\mu_{1}\boxplus\mu_{2}}(z)=G_{\mu_{1}}(Z_{1}(z))=G_{\mu_{2}}(Z_{2}(z))$ for all $z\in\mathbb{C}^{+}$. Using Speicher’s combinatorial approach [30] to freeness, Biane [11] proved this result in the general case.

Bercovici and Belinschi [6], Belinschi [7], Chistyakov and Götze [15], proved, using complex analytic methods, that there exist unique functions $Z_{1}(z)$ and $Z_{2}(z)$ in the class $\mathcal{F}$ such that, for $z\in\mathbb{C}^{+}$,

[TABLE]

The function $F_{\mu_{1}}(Z_{1}(z))$ belongs again to the class $\mathcal{F}$ and there exists $\mu\in\mathcal{M}$ such that $F_{\mu_{1}}(Z_{1}(z))=F_{\mu}(z)$, where $F_{\mu}(z)=1/G_{\mu}(z)$ and $G_{\mu}(z)$ is the Cauchy transform as in (3.3). The measure $\mu$ depends on $\mu_{1}$ and $\mu_{2}$ only and $\mu=\mu_{1}\boxplus\mu_{2}$.

Specializing to $\mu_{1}=\mu_{2}=\dots=\mu_{n}=\mu$ write $\mu_{1}\boxplus\dots\boxplus\mu_{n}=\mu^{n\boxplus}$. The relation (3.10) admits the following consequence (see for example [15], Section 2, Corollary 2.3).

Proposition 3.3**.**

Let $\mu\in\mathcal{M}$. There exists a unique function $Z\in\mathcal{F}$ such that

[TABLE]

and $F_{\mu^{n\boxplus}}(z)=F_{\mu}(Z(z))$.

Using the representation (3.1) for $F_{\mu}(z)$ we obtain

[TABLE]

where $\tau$ is a nonnegative measure such that $\tau(\mathbb{R})=\Im F_{\mu}(i)-1$. Denote $z=x+iy$, where $x,y\in\mathbb{R}$. We see that, for $\Im z>0$,

[TABLE]

For every real fixed $x$, consider the equation

[TABLE]

Since $y\mapsto I_{\mu}(x,y),\,y>0$, is positive and monotone, and decreases to [math] as $y\to\infty$, it is clear that the equation (3.13) has at most one positive solution. If such a solution exists, denote it by $y_{n}(x)$. Note that (3.13) does not have a solution $y>0$ for any given $x\in\mathbb{R}$ if and only if $I_{\mu}(x,0)\leq 1/(n-1)$. Consider the set $S:=\{x\in\mathbb{R}:I_{\mu}(x,0)\leq 1/(n-1)\}$. We put $y_{n}(x)=0$ for $x\in S$. We proved in [17], Section 3, p.13, that the curve $\gamma_{n}$ given by the equation $z=x+iy_{n}(x),\,x\in\mathbb{R}$, is continuous and simple.

Consider the open domain $\tilde{D}_{n}:=\{z=x+iy,\,x,y\in\mathbb{R}:y>y_{n}(x)\}$.

Lemma 3.4**.**

Let $Z\in\mathcal{F}$ be the solution of the equation $(\ref{3.10})$. The function $Z(z)$ maps $\mathbb{C}^{+}$ conformally onto $\tilde{D}_{n}$. Moreover the function $Z(z),\,z\in\mathbb{C}^{+}$, is continuous up to the real axis and it establishes a homeomorphism between the real axis and the curve $\gamma_{n}$.

This lemma was proved in [17] (see Lemma 3.4). The following lemma was proved as well in [17] (see Lemma 3.5).

Lemma 3.5**.**

Let $\mu$ be a probability measure such that $m_{1}(\mu)=0,m_{2}(\mu)=1$. Assume that $\int_{|u|>\sqrt{(n-1)/8}}u^{2}\,\mu(du)\leq 1/10$ for some positive integer $n\geq 10^{3}$. Then the following inequality holds

[TABLE]

where $Z\in\mathcal{F}$ is the solution of the equation $(\ref{3.10})$.

The next lemma was proved in [24] and [38].

Lemma 3.6**.**

There exists a unique probability measure $\nu$ such that such that $F_{\mu}(z)=z-G_{\nu}(z),\,z\in\mathbb{C}^{+}$, and, for every $n\geq 1$, $F_{\mu_{n}}(z)=z-G_{\nu_{n-1}\boxplus w_{t}}(z),\,z\in\mathbb{C}^{+}\cup\mathbb{R}$, where the measure $\nu_{n-1}$ is given by $d\nu_{n-1}(x)=d\nu(\sqrt{n}x)$ and $t=t(n)=(n-1)/n$.

Biane [11] gave the following bound.

Lemma 3.7**.**

Fix $t>0$ and the probability measure $\nu$. Then $|G_{\nu\boxplus w_{t}}(z)|\leq t^{-1/2},\,z\in{\mathbb{C}}^{+}\cup\mathbb{R}$.

4. Free Meixner measures

Consider the three-parameter family of probability measures $\{\mu_{a,b,d}:a\in\mathbb{R},b<1,d<1\}$ with the reciprocal Cauchy transform

[TABLE]

which we will call the free centered (i.e. with mean zero) Meixner measures. In this formula we choose the branch of the square root determined by the condition $\Im z>0$ implies $\Im(1/G_{\mu_{a,b,d}}(z))\geq 0$. These measures are counterparts of the classical measures discovered by Meixner [25]. The free Meixner type measures occurred in many places in the literature, see for example [14], [28].

Saitoh and Yoshida [28] have proved that the absolutely continuous part of the free Meixner measure $\mu_{a,b,d},a\in\mathbb{R},b<1,d<1$, is given by

[TABLE]

when $a-2\sqrt{1-d}/(1-b)\leq x\leq a+2\sqrt{1-d}/(1-b)$, where

[TABLE]

Saitoh and Yoshida proved as well that for $0\leq b<1$ the (centered) free Meixner measure $\mu_{a,b,d}$ is $\boxplus$-infinitely divisible.

As we have shown in [17], Section 4, it follows from Saitoh and Yoshida’s results that the probability measure $\mu_{a_{n},b_{n},d_{n}}$ with the parameters $a_{n},b_{n},d_{n}$ from (2.3) is $\boxplus$-infinitely divisible and it is absolutely continuous with a density of the form (4.2) where $a=a_{n},b=b_{n},d=d_{n}$ for sufficiently large $n\geq n_{1}(\mu)$.

5. Passage to measures with bounded supports

Let us assume that $\mu\in\mathcal{M}$ and $m_{4}(\mu)<\infty$. In addition let $m_{1}(\mu)=0$ and $m_{2}(\mu)=1$. By Proposition 3.3, there exists $Z(z)\in\mathcal{F}$ such that (3.11) holds, and $F_{\mu^{n\boxplus}}(z)=F_{\mu}(Z(z))$. Hence $F_{\mu_{n}}(z)=F_{\mu}(\sqrt{n}S_{n}(z))/\sqrt{n},\,z\in\mathbb{C}^{+}$, where $S_{n}(z):=Z(\sqrt{n}z)/\sqrt{n}$. Since $m_{1}(\mu)=0,\,m_{2}(\mu)=1$ and $m_{4}(\mu)<\infty$, by Proposition 3.2, we have the representation

[TABLE]

where $\tau$ is a nonnegative measure such that $\tau(\mathbb{R})=1$ and $m_{2}(\tau)<\infty$.

Denote, for $n\in\mathbb{N}$,

[TABLE]

It is easy to see that $0<\eta(n;\tau)\leq 11$ and $\eta(n;\tau)\to 0$ monotonically as $n\to\infty$. Let $\delta_{n}\in(0,10^{-1/2}]$ be a point at which the infimum of the function $g_{n}(\varepsilon;\tau)$ is attained. This means that

[TABLE]

Consider a function

[TABLE]

This function belongs to the class $\Cal{F}$ and therefore there exists the probability measure $\mu^{*}$ such that $F_{\mu^{*}}(z)=F(z),\,z\in\mathbb{R}$. The probability measure $\mu^{*}$ of course depends on $n$. Moreover we conclude from the inversion formula that $\mu^{*}([-\frac{\sqrt{10}}{3}\delta_{n}\sqrt{n-1},\frac{\sqrt{10}}{3}\delta_{n}\sqrt{n-1}])=1$ for $n\geq n_{1}(\mu)$. Hence it follows that the support of $\mu^{*}$ is contained in the interval $[-\frac{1}{3}\sqrt{n-1},\frac{1}{3}\sqrt{n-1}]$. By Proposition 3.2, we see as well that $m_{1}(\mu^{*})=0$ and

[TABLE]

Moreover

[TABLE]

In the same way

[TABLE]

Here $m_{k}(\tau^{*}),\,k=0,1,2$, denote moments of the measure $\tau^{*}$.

Let $X^{*},X_{1}^{*},X_{2}^{*},\dots$ be free identically distributed random variables such that $\Cal{L}(X^{*})=\mu^{*}$. Denote $\mu_{n}^{*}:=\Cal{L}((X_{1}^{*}+\dots+X_{n}^{*})/\sqrt{n})$. As before, by Proposition 3.3, there exists $W(z)\in\mathcal{F}$ such that (3.11) holds with $Z=W$ and $\mu=\mu^{*}$, and $F_{(\mu^{*})^{n\boxplus}}(z)=F_{\mu^{*}}(W(z))$. Hence $F_{\mu_{n}^{*}}(z)=F_{\mu^{*}}(\sqrt{n}T_{n}(z))/\sqrt{n},\,z\in\mathbb{C}^{+}$, where $T_{n}(z):=W(\sqrt{n}z)/\sqrt{n}$. In the sequel we shall need more detailed information about the behaviour of the functions $T_{n}(z)$ and $S_{n}(z)$. By Lemma 3.4, these functions are continuous up to the real axis for $n\geq n_{1}(\mu)$. Their values for $z=x\in\mathbb{R}$ we denote by $T_{n}(x)$ and $S_{n}(x)$, respectively. In order to formulate the following results for $T_{n}(z)$ we introduce some notations. Denote by $M_{n}(z)$ the reciprocal Cauchy transform of the free Meixner measure $\mu_{a_{n},b_{n},d_{n}}$ with the parameters $a_{n},b_{n}$ and $d_{n}$ from (2.3), i.e.,

[TABLE]

Denote by $D_{n}$ the rectangle

[TABLE]

where $\varepsilon_{n1}:=c_{1}(\mu)(\eta(n;\tau)+1/\sqrt{n})$ and $c_{1}(\mu)>0$ is sufficiently large. In the sequel we assume that $\varepsilon_{n1}$ is always of this form.

Repeating step by the step the arguments of Section 7 (see Subsections 7.2–7.7) of our paper [19] we establish the following result.

Theorem 5.1**.**

Let $\mu\in\mathcal{M}$ such that $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$. Then there exists a constant $c(\mu)$ such that the following relation holds, for $z\in D_{n}$ and $n\geq n_{1}(\mu)$,

[TABLE]

In addition

[TABLE]

In (5.7) and (5.8) $\theta$ is a complex-valued quantities such that $|\theta|\leq 1$.

Here and in the sequel constants $c(\mu)>0$ do not depend on the constant $c_{1}(\mu)$.

In Section 7 of this paper we shall give a more detailed exposition of the proof of this theorem.

By Lemmas 3.4, 3.5, $|T_{n}(z)|\geq 1.03/3$ for $z\in\mathbb{C}^{+}\cup\mathbb{R}$ and for $n\geq n_{1}(\mu)$. It is obvious that the same estimate holds for $M_{n}(z)$. Since

[TABLE]

we conclude that $G_{\mu_{n}^{*}}(z)$ is a continuous function up to the real axis. Denote its value for real $x$ by $G_{\mu_{n}^{*}}(x)$. Denote $G_{\hat{\mu}_{n}}(z):=1/T_{n}(z),\,z\in\mathbb{C}^{+}$. This function is continuous up to the real axis as well. Therefore $\hat{\mu}_{n}$ and $\mu_{n}^{*}$ are absolutely continuous measures with continuous densities $\hat{p}_{n}(x)$ and $p_{n}^{*}(x)$, respectively,

[TABLE]

In addition, $\hat{p}_{n}(x)\leq 1$ and $p_{n}^{*}(x)\leq 50$ for all $x\in\mathbb{R}$ and $n\geq n_{1}(\mu)$.

Theorem 5.2**.**

Let $\mu\in\mathcal{M}$ such that $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$. Then, for $x\in I_{n}:=\{x\in\mathbb{R}:|x-a_{n}|\leq\frac{2}{e_{n}}-\frac{\varepsilon_{n1}}{n}\}$ and $n\geq n_{1}(\mu)$, the following relation holds

[TABLE]

where $v_{n}(x)$ is defined in $(\ref{vden})$.

Proof.

We shall use the following estimate, for $x\in\mathbb{R}$,

[TABLE]

where

[TABLE]

By (5.7) and by the lower bounds

[TABLE]

we easily obtain the upper bound, for $x\in I_{n}$ and $n\geq n_{1}(\mu)$,

[TABLE]

and, by (5.8), we have

[TABLE]

Since, by (4.2),

[TABLE]

we easily conclude that

[TABLE]

for $x\in\tilde{I}_{n}$. Using again (5.7) and (5.13), we obtain

[TABLE]

On the other hand it is not difficult to show that

[TABLE]

which leads to the relation

[TABLE]

for $x\in\tilde{I}_{n}$.

Applying (5.14), (5.15), (5) and (5.17), (5.19) to (5) we arrive at the statement of the theorem. ∎

6. Local asymptotic expansion

First we prove the auxiliary result.

Theorem 6.1**.**

Let $\mu\in\mathcal{M}$ such that $m_{4}(\mu)<\infty$ and $m_{1}(\mu)=0,\,m_{2}(\mu)=1$. Then the following relation holds

[TABLE]

where

[TABLE]

and $\tilde{\rho}_{n2}(x)$ is a continuous function such that

[TABLE]

Proof.

Represent the density $p_{n}(x)$ of the measure $\mu_{n}$ in the form

[TABLE]

where $p_{nj}(x)\geq 0,\,x\in\mathbb{R},\,j=1,2$, and, for $z\in\mathbb{C}^{+}$,

[TABLE]

Since $\lim_{y\to+\infty}iyI_{2}(iy)=\int_{|u|>\sqrt{n-1}/3}\mu(du)=\int_{\mathbb{R}}p_{n2}(u)\,du$, we note that

[TABLE]

Since $S_{n}(x),\,x\in\mathbb{R}$, is a continuous function and $|S_{n}(x)|\geq 1.03/3$ for all $x\in\mathbb{R}$, we easily see that, for $x\in\mathbb{R}$,

[TABLE]

and that $p_{n1}(x)$ is a continuous function on the real line. In view of (2.2), $p_{n}(x)$ is a continuous function on the real line and $p_{n}(x)\leq c(\mu),\,x\in\mathbb{R}$, for $n\geq n_{1}(\mu)$. Therefore we conclude from (6.2) that $p_{n2}(x)$ is a continuous function on the real line and $p_{n2}(x)\leq c(\mu),\,x\in\mathbb{R}$, for the same $n$.

Now we may write

[TABLE]

Since

[TABLE]

we have

[TABLE]

By the inequalities $|S_{n}(x)-u/\sqrt{n}|\geq 0.01$, $|T_{n}(x)-u/\sqrt{n}|\geq 0.01$ for $x\in\mathbb{R}$ and $|u|\leq\sqrt{n-1}/3$, we conclude that, for $x\in\mathbb{R}$,

[TABLE]

On the other hand we note that, for $x\in\mathbb{R}$,

[TABLE]

Using (5)–(5) and the inequality

[TABLE]

we obtain the estimate

[TABLE]

Applying (6.5) and (6.6) to (6.4), we have, for $x\in\mathbb{R}$,

[TABLE]

The representation (6.1) follows immediately from (6.2) if to define $\tilde{\rho}_{n1}(x)=p_{n1}(x)-p^{*}_{n}(x)$ and $\tilde{\rho}_{n2}(x)=p_{n2}(x)$ and from the bounds (6.7) and (6.3). ∎

7. Proof of Theorem 5.1

In this section we show how the arguments of Section 7 in [19] lead to a proof of Theorem 5.1.

Repeating the arguments of Subsection 7.2 we deduce that $T_{n}(z)$ satisfies the functional equation, for $z\in\mathbb{C}^{+}$,

[TABLE]

where $\zeta_{n1}(z):=\int_{\mathbb{R}}\frac{u^{5}\,\mu^{*}(du)}{W(\sqrt{n}z)-u},$ $\zeta_{n2}(z):=m_{3}(\mu^{*})-z/\sqrt{n}$, $\zeta_{n3}(z)(z):=m_{4}(\mu^{*})+\zeta_{n1}(z)-zm_{3}(\mu^{*})/\sqrt{n}$ and $\zeta_{n4}(z)(z):=m_{4}(\mu^{*})+\zeta_{n1}(z)$. As in Subsection 7.3 from [19] we obtain estimates for the functions $\zeta_{nj}(z),j=1,2,3,4$, in the domain $D^{*}:=\{|\Re z|\leq 4,\,0<\Im z\leq 3\}$

[TABLE]

For every fixed $z\in\mathbb{C}^{+}$ consider the equation

[TABLE]

Denote the roots of the equation (7.3) by $w_{j}=w_{j}(z),\,j=1,\dots,5$.

As in Subsection 7.4 [19] we can show that for every fixed $z\in D^{*}$ the equation $Q(z,w)=0$ has three roots, say $w_{j}=w_{j}(z),\,j=1,2,3$, such that

[TABLE]

and two roots, say $w_{j},\,j=4,5$, such that $|w_{j}|\geq r^{\prime}$ for $j=4,5$.

Represent $Q(z,w)$ in the form

[TABLE]

where $w^{3}+g_{1}w^{2}+g_{2}w+g_{3}=(w-w_{1})(w-w_{2})(w-w_{3})$. From this formula we derive the relations

[TABLE]

By Vieta’s formulae and (7.4), note that

[TABLE]

Now we obtain from (7) and (7.6) the following bounds, for $z\in D^{*}$,

[TABLE]

Then we conclude from (5), (7.2), (7)–(7.7) that, for the same $z$,

[TABLE]

Now repeating the arguments of Subsection 7.4 we deduce the inequality

[TABLE]

To find the roots $w_{4}$ and $w_{5}$, we need to solve the equation $w^{2}+s_{1}w+s_{2}=0$. Using (7), we have, for $j=4,5$,

[TABLE]

where

[TABLE]

The quantities $r_{n1}(z)$ and $r_{n2}(z)$ admit the bound (see Subsection 7.4 and 7.5 from [19])

[TABLE]

We choose the branch of the analytic square root according to the condition $\Im w_{4}(i)\geq 0$.

As in Subsection 7.6 from [19] we prove that $w_{4}(z)=T_{n}(z)$ for $z\in D_{n}$, where the constant $c(\mu)$ in (7.11) does not depend on the constant $c_{1}(\mu)$. Since the constant $c_{1}(\mu)$ is sufficiently large, we have, by (7.11),

[TABLE]

For $z\in D_{n}$, using formula (7) with $j=4$ for $T_{n}(z)$, we write

[TABLE]

Using (7.12) and the power expansion for the function $(1+z)^{1/2},\,|z|<1$, we easily rewrite (7) in the form

[TABLE]

where $|r_{n3}(z)|\leq c(\mu)\eta(n;\tau)/n$. The relation (5.7) immediately follows from (7.14).

Using (5.10), we conclude that

[TABLE]

Since $|T_{n}(z)|\geq 1.03/3$ for $z\in\mathbb{C}^{+}$ and $n\geq n_{1}(\mu)$, we arrive at (5.8).

The function $T_{n}(x)$ for real $x$ such that $\frac{2}{e_{n}}-\frac{\varepsilon_{n1}}{n}\leq|x-a_{n}|\leq 3$ coincide with $w_{3}(x)$ or $w_{4}(x)$ from (7). Here we understand $w_{j}(x)$ as limit values of $w_{j}(z)$ where $z\in D^{*}$ and $z\to x$. It is not difficult to conclude from the formula (7) that $0\leq\Im T_{n}(x)\leq c(\mu)\sqrt{\frac{\varepsilon_{n1}}{n}}$ for $\frac{2}{e_{n}}-\frac{\varepsilon_{n1}}{n}\leq|x-a_{n}|\leq 3$.

8. Proof of Theorem 2.1

Recalling the definition of the function $Z(z)$, we see, by Lemma 3.4, that the function $S_{n}(z)$ maps $\mathbb{C}^{+}$ conformally onto $\hat{D}_{n}$, where $\hat{D}_{n}:=\{z=x+iy,x,y\in\mathbb{R}:y>y_{n}(x\sqrt{n})/\sqrt{n}\}$. Denote by $\hat{\gamma}_{n}$ a curve given by the equation $z=x+i\hat{y}_{n}(x)$, where $\hat{y}_{n}(x)=y_{n}(x\sqrt{n})/\sqrt{n}$. The function $S_{n}(z)$ is continuous up to the real axis and it establishes a homeomorphism between the real axis and the curve $\hat{\gamma}_{n}$.

Note as well that the function $T_{n}(z)$ maps $\mathbb{C}^{+}$ conformally onto $\hat{D}_{n}^{*}$, where $\hat{D}_{n}^{*}:=\{z=x+iy,x,y\in\mathbb{R}:y>y_{n}^{*}(x\sqrt{n})/\sqrt{n}\}$. Here $y_{n}^{*}(x)$ is defined in the same way as $y_{n}(x)$ if we change the measure $\mu$ by $\mu^{*}$. By definition of the measure $\mu^{*}$, we see that $y_{n}^{*}(x)\leq y_{n}(x)$. Hence $S_{n}(\mathbb{C}^{+})\subseteq T_{n}(\mathbb{C}^{+})$. Denote by $\hat{\gamma}_{n}^{*}$ a curve given by the equation $z=x+i\hat{y}_{n}^{*}(x),\,x\in\mathbb{R}$, where $\hat{y}_{n}^{*}(x)=y_{n}^{*}(x\sqrt{n})/\sqrt{n}$. The function $T_{n}(z)$ is continuous up to the real axis and it establishes a homeomorphism between the real axis and the curve $\hat{\gamma}_{n}^{*}$.

Let $x\in\mathbb{R}$. Since $S_{n}(z)$ and $T_{n}(z)$ are the conformal maps $\mathbb{C}^{+}$ on $\hat{D}_{n}$ and $\hat{D}_{n}^{*}$, respectively, which are continuous up to the real axis, we note that the functions $\Re S_{n}(x)$ and $\Re T_{n}(x)$ are monotonically increasing. Hence for every $x\in\mathbb{R}$ there exists unique $\tilde{x}\in\mathbb{R}$ such that $\Re S_{n}(x)=\Re T_{n}(\tilde{x})$. Denote $h(x):=\Im S_{n}(x)-\Im T_{n}(\tilde{x})\geq 0$.

In order to prove Theorem 2.1 we need the following auxiliary results. In the sequel we assume that $n\geq n_{1}(\mu)\geq 10$.

Proposition 8.1**.**

For $z_{1},z_{2}\in T_{n}(\mathbb{C}^{+})$,

[TABLE]

Proof.

Using the formula

[TABLE]

we have the relation, for $z_{1},z_{2}\in\mathbb{C}^{+}$,

[TABLE]

Since, by Lemma 3.5, $|u-z\sqrt{n}|\geq 10^{-2}\sqrt{n}$ for $|u|\leq\sqrt{n-1}/\pi$ and $z\in T_{n}(\mathbb{C}^{+})$, we immediately arrive at the assertion of the proposition. ∎

Proposition 8.2**.**

For $z\in S_{n}(\mathbb{C}^{+})$,

[TABLE]

Proof.

Using (8.1) and the formula

[TABLE]

we have, taking into account that $S_{n}(\mathbb{C}^{+})\subseteq T_{n}(\mathbb{C}^{+})$,

[TABLE]

for $z\in S_{n}(\mathbb{C}^{+})$, proving the proposition. ∎

Proposition 8.3**.**

For $x_{1},x_{2}\in I_{n}$, we have the estimate

[TABLE]

Proof.

By Theorem 5.1, we have the following relation

[TABLE]

where $x_{1},x_{2}\in I_{n}$. On the other hand it is easy to see that

[TABLE]

Moreover, we have, for $x_{1},x_{2}\in I_{n}$,

[TABLE]

In view of this relation and (8), (8.4), we easily obtain the assertion of the proposition. ∎

Proposition 8.4**.**

For $x\in I_{n}$, the following formula holds

[TABLE]

Proof.

The proof immediately follows from Theorem 5.1. ∎

Proposition 8.5**.**

For $x\in\mathbb{R}$, the following estimates hold

[TABLE]

Proof.

We note from (3.11) that $S_{n}(z)=\frac{z}{n}+\frac{n-1}{n}F_{\mu_{n}}(z),$ where $\mu_{n}$ is the distribution of $Y_{n}$. By Proposition 3.6, we see that $S_{n}(z)=z-G_{\nu_{n-1}\boxplus w_{t}}(z),\,z\in\mathbb{C}^{+}\cup\mathbb{R}$, where the measure $\nu_{n-1}$ is given by $d\nu_{n-1}(x)=d\nu(\sqrt{n}x)$, where, by (5.1), $\nu=\tau$, and $t=t(n)=(n-1)/n$. Since, by Proposition 3.7, $|G_{\nu_{n-1}\boxplus w_{t}}(z)|\leq\sqrt{\frac{n}{n-1}},\,z\in\mathbb{C}^{+}\cup\mathbb{R}$, we obtain the upper bound (8.5).

Now we note that in the same way as above $T_{n}(z)=z-G_{\zeta_{n-1}\boxplus w_{t}}(z),\,z\in\mathbb{C}^{+}\cup\mathbb{R}$, where the measure $\zeta_{n-1}$ is given by $d\zeta_{n-1}(x)=d\zeta(\sqrt{n}x)$, where, by (5.3), $\zeta$ a narrowing of the measure $\tau$ on the interval $[-\delta\sqrt{n-1},\delta\sqrt{n-1}]$. Moreover $|G_{\zeta_{n-1}\boxplus w_{t}}(z)|\leq\sqrt{\frac{n}{n-1}},\,z\in\mathbb{C}^{+}\cup\mathbb{R}$.

It remains to write the following relation

[TABLE]

and obtain from here the upper bound (8.6). The proposition is proved. ∎

Proposition 8.6**.**

Let $x\in\mathbb{R}$. Then

[TABLE]

Proof.

Without loss of generality we assume that $\Im T_{n}(\tilde{x})>0$. The case $\Im T_{n}(\tilde{x})=0$ considers in the same way. It follows from (8.1) that

[TABLE]

The formula (8.2) gives us

[TABLE]

For $x\in\mathbb{R}$ we have the following lower bound

[TABLE]

and the upper bound

[TABLE]

It follows from (8.8) and (8.9) that $\tilde{I}_{1}=\tilde{I}_{2}$, therefore we obtain the assertion of the proposition from (8) and (8.11). ∎

Proposition 8.7**.**

Let $x\in\mathbb{R}$. Then

[TABLE]

Proof.

Without loss of generality we assume that $\Im S_{n}(x)>0$. By the formula (8.1), we have

[TABLE]

On the other hand, by (8.2), we obtain

[TABLE]

It follows from these formulae that

[TABLE]

where

[TABLE]

and

[TABLE]

It is easy to see that

[TABLE]

Therefore, using (8.7), we have

[TABLE]

and (8.12) is proved. ∎

Proposition 8.8**.**

For $x\in I_{n}^{*}$, the following inequalities hold

[TABLE]

Proof.

Consider $x$ such that $|\tilde{x}-a_{n}|\leq\frac{2}{e_{n}}-\sqrt{\frac{\varepsilon_{n1}}{c_{1}(\mu)n}}$. By (8.5) and (8.6) we have $|x-\tilde{x}|\leq 3$ and $|S_{n}(x)|\leq 7$. In view of Proposition 8.4 and (5), it is easy to see that

[TABLE]

By (8.12), we see that

[TABLE]

Since $c(\mu)$ does not depend on $c_{1}(\mu)$, we conclude finally

[TABLE]

The function $x(\tilde{x})$ is monotone, continuous and the assertion of the proposition follows at once from (8.17). ∎

Proposition 8.9**.**

We have the bounds

[TABLE]

Proof.

Let $x\in I_{n}^{*}$. We have the formula

[TABLE]

By Propostion 8.8, if $x\in I_{n}^{*}$, then $\tilde{x}\in I_{n}$. We see, by Propostion 8.4, that, for such $x$,

[TABLE]

Using again Proposition 8.8 we obtain

[TABLE]

By Propositions 8.6– 8.8, and the formula (8.20), we conclude that

[TABLE]

Therefore, using (8.22), we get

[TABLE]

Applying (8.21) and (8) to (8.19) we arrive at the assertion of the proposition. ∎

Proposition 8.10**.**

We have the bounds

[TABLE]

Proof.

Since $\Re S_{n}(x)-\Re T_{n}(x)=\Re T_{n}(\tilde{x})-\Re T_{n}(x)$, we conclude, using Proposition 8.4 and (8.22),

[TABLE]

for $x\in I_{n}^{*}$. The proposition is proved. ∎

Proposition 8.11**.**

For $x\in I_{n}\setminus I_{n}^{*}$ and $\alpha\in(0,1]$, we have

[TABLE]

Proof.

Let $x,\tilde{x}\in I_{n}\setminus I_{n}^{*}$ and $\alpha\in(0,1]$. We have the two possibilities

[TABLE]

Consider the case $a)$. Then, by (8.7), we have

[TABLE]

In addition, repeating the argument of the proof of Proposition 8.9 and using the inequality $|\tilde{x}-x|\leq\sqrt{\frac{\varepsilon_{n1}}{n}}$, we obtain

[TABLE]

Hence in the case $a)$

[TABLE]

Now consider the case $b)$. In this case we have the following simple estimate

[TABLE]

It remains to consider the case when $x\in I_{n}\setminus I_{n}^{*}$ and $\tilde{x}\not\in I_{n}\setminus I_{n}^{*}$.

Let $x_{n2}^{*}=a_{n}+2/e_{n}-\sqrt{\frac{\varepsilon_{n1}}{n}}$ and $x_{n2}=a_{n}+2/e_{n}-\frac{\varepsilon_{n1}}{n}$. Assume that $x_{n2}^{*}<x\leq x_{n2}$ and $\tilde{x}\in I_{n}^{*}$. Assume as well that $a)$ holds. By (8.12), we see that $|\tilde{x}(x_{n2}^{*})-x_{n2}^{*}|\leq c(\mu)\sqrt{\frac{\varepsilon_{n1}}{n}}$. In addition $\tilde{x}(x)$ is a monotone increasing function, therefore $|\tilde{x}(x)-x|\leq c(\mu)\sqrt{\frac{\varepsilon_{n1}}{n}}$. Repeating the previous estimates we obtain the bounds (8.27) and (8.28) in the considered case. We prove the bounds (8.27) and (8.28) for $x_{n1}<x\leq x_{n1}^{*}$, where $x_{n1}^{*}=a_{n}-2/e_{n}+\sqrt{\frac{\varepsilon_{n1}}{n}},\,x_{n1}=a_{n}-2/e_{n}+\frac{\varepsilon_{n1}}{n}$ and $\tilde{x}\in I_{n}^{*}$ in the same way.

Without loss of generality let us assume now that $\tilde{x}>x_{n2}$. Then, by (8.12) and (8.20), we have

[TABLE]

Then, by (5.9), we conclude $\Im T_{n}(\tilde{x})\leq c(\mu)\sqrt{\frac{\varepsilon_{n1}}{n}}$ and hence

[TABLE]

Since in our case (8.26) holds, we arrive at the upper bound

[TABLE]

In the case $b)$ we have obviously the estimate (8.28). The proposition is proved. ∎

Proposition 8.12**.**

For $x\in I_{n}\setminus I_{n}^{*}$,

[TABLE]

Proof.

Note that, for every fixed $x\in I_{n}\setminus I_{n}^{*}$,

[TABLE]

for

[TABLE]

Moreover, for this $\alpha$,

[TABLE]

Therefore the assertion of the proposition follows immediately from Proposition 8.11. ∎

Now we finish the proof of Theorem 2.1. Return to the formulations of Theorem 5.2 and 6.1. Denote

[TABLE]

for $x\in I_{n}$. The statement of Theorem 2.1 for $x\in I_{n}$ follows immediately from Propositions 8.9– 8.10 and 8.12.

It remains to prove (2.10). From Theorem 2.6 [19] and the formula (5.17) it follows immediately that

[TABLE]

where $\varepsilon_{n2}:=\eta_{q}(n)\frac{\beta_{q}(\mu)}{n^{(q-4)/2}}$ if $\beta_{q}(\mu):=\int_{\mathbb{R}}|u|^{q}\,\mu(du)<\infty,\,4\leq q<5$. Here

[TABLE]

with $q_{*}=5-\min\{q,5\}$. It is easy to see that $\eta_{q}(n)$ are the functions such that $\eta_{q}(n)\leq 10^{1+3/2}$ and $\eta_{q}(n)\to 0$ monotonically as $n\to\infty$. If $\beta_{q}(\mu)<\infty,\,q\geq 5$, then $\varepsilon_{n2}:=\frac{\beta_{5}(\mu)}{n^{1/2}}$. Therefore (2.10) holds with the indicated sequence $\{\varepsilon_{n2}\}$.

Theorem 2.1 is completely proved.

Remark 8.13*.*

It is obvious that for the considered above $\varepsilon_{n2}$ we have the properties: $\frac{\varepsilon_{n2}}{n}=o(1/n^{(q-2)/2})$ if $\beta_{q}(\mu)<\infty,\,4\leq q<5$, and $\frac{\varepsilon_{n2}}{n}=O(1/n^{3/2})$ if $\beta_{5}(\mu)<\infty$.

Remark 8.14*.*

If $m_{2k}(\mu)<\infty$ for some $k=3,\dots$, then we choose in (5.2) $\delta_{n}=1/\pi$ and define $\varepsilon_{n1}:=c_{1}(\mu)(n\tau(\mathbb{R}\setminus[-\sqrt{n-1}/\pi,\sqrt{n-1}/\pi])+1/\sqrt{n})$. In this case $\varepsilon_{n1}=O(1/\sqrt{n})$. Repeating the argument of Sections 5–8 we obtain the statement of Theorem 2.1 with such $\varepsilon_{n1}$ and $\varepsilon_{n2}$ described before.

9. Asymptotic expansion of $\int_{\mathbb{R}}|p_{n}(x)-p_{w}(x)|\,dx$

In this section we shall prove Corollary 2.2. Indeed, by the estimate (2.10), we have, for $n\geq n_{1}$,

[TABLE]

Using Theorems 2.1, we easily conclude that

[TABLE]

By (2.7), we see that

[TABLE]

and, by (2.8),

[TABLE]

Applying these relations to (9.1) we get the expansion (2.11).

10. Asymptotic expansion of the free entropy

In this section we prove Corollaries 2.3. First we find an asymptotic expansion of the logarithmic energy $E(\mu_{n})$ of the measure $\mu_{n}$. Recall that (see [20])

[TABLE]

Using (2.2) and the equality $\int_{\mathbb{R}}u^{2}\,p_{n}(u)\,du=1$, we get the inequality

[TABLE]

Therefore we conclude with the help of the Cauchy-Bunyakovsky inequality that

[TABLE]

Recalling (2.10), we obtain

[TABLE]

Now we note that

[TABLE]

Using the form of $v_{n}(x)$ we easily conclude that

[TABLE]

Recalling the definition of $v_{n}(x)$ we see that

[TABLE]

In view of $E(\mu_{w})=1/4$ and $e_{n}^{-2}=1-d_{n}+2b_{n}+c(\mu)\theta n^{-2},\,\log e_{n}=\frac{d_{n}}{2}-b_{n}+c(\mu)\theta n^{-2}$, note that

[TABLE]

Since the function $\int_{\mathbb{R}}\log|x-y|p_{w}(y)\,dy$ is even, we see that $\tilde{I}_{2}(v_{n})=0$. In order to calculate $\tilde{I}_{3}(v_{n})$ we easily deduce that

[TABLE]

Therefore we obtain

[TABLE]

Using the following well-known formula (see [20], p. 197)

[TABLE]

we deduce

[TABLE]

Therefore

[TABLE]

By (10.4)–(10.11), we arrive at the formula

[TABLE]

Now we note, using (10.7), (10.9), (10.10), and the Hölder inequality that, for $x\in I_{n}-a_{n}$ and for any fixed positive $\delta$,

[TABLE]

where $c(\mu,\delta)$ are positive constants depending on $\mu$ and $\delta$ only.

On the other hand, by (8.29), we see that

[TABLE]

The last two relations give us finally

[TABLE]

It remains to estimate $I_{13}(\mu_{n})$. Write

[TABLE]

In order to estimate $I_{13,1}(\mu_{n})$ we use the upper bound

[TABLE]

Let $p,q>1$ and $\frac{1}{p}+\frac{1}{q}=1$. We assume that $q$ is closed to $1$, i.e., $1<q\leq 1.01$. By Hölder inequality, for $x\in I_{n}^{*}$,

[TABLE]

and, by (2.9),

[TABLE]

where $c(\mu,q)>0$ is a constant depended on $\mu$ and $q$ only. From the two last upper bounds we get

[TABLE]

On the other hand it is easy to see that

[TABLE]

We conclude from (10.15) and (10.16) that

[TABLE]

Now introduce the quantity

[TABLE]

As above we obtain, for $x\in I_{n}$,

[TABLE]

Therefore we have

[TABLE]

We deduce from (10.16) and (10.18)

[TABLE]

It remains to estimate $I_{13,3}(\mu_{n})$. It is easy to verify that

[TABLE]

We obtain from (10.18) and (10.20) that

[TABLE]

It remains to note that the upper bound

[TABLE]

follows immediately from (10.17), (10.19), and (10.21).

In view of (10), (10.2), (10.12), (10.13) and (10.22) we get

[TABLE]

The assertion of Corollary 2.3 follows from this relation.

11. Asymptotic expansion of the free Fisher information

Now let us prove Corollary 2.4. We shall show that the free Fisher information of the measure $\mu_{n}$ has the form (2.13). Denote

[TABLE]

As before we see that, by (2.10),

[TABLE]

On the other hand, by (2.6), we have

[TABLE]

In the sequal we consider nonnegative entire numbers $0\leq k\leq 2,\,l\geq 1$ and $k+l=3$ only.

We see, by (2.9), that

[TABLE]

Now we note, by (2.7), that

[TABLE]

Furthermore, we have, using the bound (2.8),

[TABLE]

From (11) and (11) it follows

[TABLE]

Applying (11.4) and (11.7) to (11.3) we obtain

[TABLE]

It is easy to see that the integral on the right hand-side of (11.8) is equal to

[TABLE]

Therefore we finally conclude by (11.1)–(11.3) that

[TABLE]

Thus, Corollary 2.4 is proved.