Readership: Graduate students and researchers interested in random matrices.

Chapters: 1. Circulants, 2. Symmetric and reverse circulant, 3. LSD: normal approximation, 4. LSD: dependent input, 5. Spectral radius: light tail, 6. Spectral radius: k‐circulant, 7. Maximum of scaled eigenvalues: dependent input, 8. Poisson convergence, 9. Heavy‐tailed input: LSD, 10. Heavy‐tailed input: spectral radius, 11. Appendix.

A k‐circulant A _k,n (1 = k = n– 1) is an n‐square matrix whose each row is obtained from the previous by k circular shifts to the right. Its first row (x ₀, …, x _{n− 1}) is called the input. Nothing is said about the x _i's in the definitions, but I guess that they are real numbers. The matrix A _k,n is a circulant C _n if k = 1 and a reverse circulant RC _n if k = n – 1. (The term ‘k‐circulant’ has also another meaning. It is often defined that a matrix is a k‐circulant if it is obtained from C _n by multiplying with k its all entries below the main diagonal.) If the sequence (x ₁, …, x _{n −1}) is palindromic, then C _n is symmetric, denoted by SC _n. Also, RC _n is symmetric.

The empirical spectral distribution function (ESD) of an n‐square matrix is obtained by putting a mass 1/n at its each eigenvalue. The limiting spectral distribution function (LSD) of a sequence of n‐square matrices is the weak limit (if it exists) of the sequence of their ESDs. In the case of random matrices, this limit is understood in some probabilistic sense.

A description of some main points of Chapters 1–8 and 11 follows. The titles of Chapters 9 and 10 are enough informative.

Chapter 1. The matrices A _k,n, C _n, RC _n, and SC _n are defined. Call them ‘circulant‐type matrices’. A formula for the eigenvalues of A _k,n is given. This formula is fundamental in what follows (except Chapter 2).

Chapter 2. The ESD and LSD are defined. A general technique to find the LSD of symmetric random matrices, based on the moment method, is introduced and applied to RC _n and SC _n when the input is i.i.d. (i.e. it consists of independent and identically distributed random variables).

Chapter 3. Using normal approximation, the LSD of C _n and, more generally, of A _k,n (for certain pairs k,n) with independent input is obtained.

Chapter 4. The results of Chapters 2 and 3 are extended to the case when the input follows a stationary linear process.

Chapter 5. Using a sharper normal approximation, the limiting behaviour of the spectral radius of C _n, RC _n, and SC _n is studied when the input is i.i.d.

Chapter 6. The results of Chapter 5 are extended to A _k,n when n = kspidummy^g+ 1.

Chapter 7. The results of Chapters 5 and 6 are extended to the case when the input is as in Chapter 4.

Chapter 8. The joint behaviour of the eigenvalues of random circulant‐type matrices with light‐tailed i.i.d. input is studied via the point process approach.

Chapter 11. The formula for the eigenvalues of A _k,n is proved. Certain notion and results in probability theory are summarised. Three auxiliary theorems, used repeatedly, are presented.

This book seems to be a useful ‘state‐of‐the‐art’ on random circulant‐type matrices. The authors are experts in this field, and several results are due to them. The reader is supposed to have much knowledge in advanced probability theory. Chapter 11 helps him or her if needed in this regard. The exercise sections, completing Chapters 1–10, are instructive, but they would have become more instructive if solutions or hints to the problems had been given.

What is the motivation of a book on random circulants?

The preface states: ‘Circulant matrices have been around for a long time and have been extensively used in many scientific areas. The classic book Circulant Matricesby P. Davis, has a wealth of information on these matrices. New research on, and applications of, these matrices are continually appearing everywhere’. This is repeated in the introduction, and some applications are mentioned. For example (p. xv), the periodogram of a sequence is a function of the eigenvalues of a suitable circulant. But all these are applications of non‐random circulants.

As far as applications of random circulants (in fact, of random matrices) are concerned, the authors only say (p. xvi) that the behaviour of the eigenvalues of such matrices with large dimension ‘has attracted considerable interest in physics, mathematics, statistics, wireless communication and other branches of sciences’. Certain important books on random matrices do not appear in the bibliography.

In the beginning of Chapter 1, the applications, already mentioned in the introduction, are repeated. In the beginning of Chapters 2–10, no applications are mentioned. Neither did I find them elsewhere.

A book on random circulants that would become a classic (as Davis' book is on non‐random circulants) is, regardless of the merits of the book under review, yet to be written. Such a book would contain applications and would also provide a comprehensive survey that ties the theory of random circulant‐type matrices to the more general theory of patterned random matrices and, still more generally, to the theory of random matrices.

读者群：对随机矩阵感兴趣的研究生和研究人员。

章节：1.循环量，2.对称和反向循环量，3. LSD：正态近似，4. LSD：从属输入，5.光谱半径：轻尾，6.光谱半径：k循环，7.标定特征值的最大值：相关输入，8.泊松收敛，9.重尾输入：LSD，10.重尾输入：频谱半径，11.附录。

一个k循环变量A _k，n（1 = k = n – 1）是一个n平方矩阵，其每行都是从前一个矩阵向右右移位k个圆而获得的。它的第一行（x ₀，…，x _{n − 1}）称为输入。关于定义中的x _i，我什么都没说，但我想它们是实数。如果k = 1，则矩阵A _k，n是循环量C _n，而反向循环量RC _n 如果k = n –1。（术语“ k循环”也具有另一种含义。通常定义为矩阵是k循环，如果它是从C _n通过将其所有在主对角线以下的项乘以k而获得的）。）如果序列（x ₁，…，x _{n -1}）是回文的，则C _n是对称的，用SC _n表示。另外，RC _n是对称的。

n平方矩阵的经验光谱分布函数（ESD）是通过将质量1 / n置于每个特征值来获得的。n平方矩阵序列的极限频谱分布函数（LSD）是其ESD序列的弱极限（如果存在）。对于随机矩阵，从某种意义上可以理解此限制。

以下是第1-8章和第11章的一些要点的描述。第9章和第10章的标题提供了足够的信息。

第1章定义了矩阵A _k，n，C _n，RC _n和SC _n。称它们为“循环型矩阵”。给出了A _k，n特征值的公式。接下来的内容（第2章除外）是该公式的基础。

第2章定义了ESD和LSD。引入了一种基于矩量法的对称对称矩阵的LSD查找通用技术，并将其应用于输入iid（即，由独立且均匀分布的随机变量组成）时的RC _n和SC _n。

第3章。使用正态逼近，获得C _n的LSD ，更一般而言，获得具有独立输入的A _k，n（对于某些对k，n）的LSD 。

第4章第2章和第3章的结果扩展到输入遵循平稳线性过程的情况。

第5章。使用更尖锐的正态近似，研究了当输入为id时C _n，RC _n和SC _n的光谱半径的极限行为。

第6章当n = k spidummy ^g + 1时，第5章的结果扩展为A _k，n。

第7章将第5章和第6章的结果扩展到第4章输入的情况。

第8章，通过点过程方法研究了带有轻尾iid输入的随机循环型矩阵特征值的联合行为。

第11章_，证明了A _k，n特征值的公式。总结了概率论中的某些概念和结果。提出了三个反复使用的辅助定理。

这本书似乎是关于随机循环类型矩阵的有用的“最新技术”。作者是该领域的专家，因此有一些结果。读者应该对高级概率论有很多了解。如果需要，第11章将为他或她提供帮助。练习部分（完成第1-10章）具有指导意义，但如果已提出解决方案或对问题的提示，它们将变得更有指导意义。

关于随机循环量的书的动机是什么？

序言指出：“循环矩阵已经存在很长时间了，并且在许多科学领域得到了广泛的应用。戴维斯（P. Davis）撰写的经典著作《循环矩阵》（Circulant Matrices）具有有关这些矩阵的大量信息。这些矩阵的新研究和应用不断出现在各个地方。在引言中重复此过程，并提及一些应用程序。例如（p。xv），序列的周期图是合适循环量特征值的函数。但是所有这些都是非随机循环的应用。

至于随机循环量（实际上是随机矩阵）的应用，作者只说（第xvi页），这类大尺寸矩阵的特征值的行为引起了物理学，数学，统计学的极大兴趣。 ，无线通信和其他科学领域”。某些有关随机矩阵的重要书籍未出现在参考书目中。

在第1章的开头，重复了引言中已经提到的应用程序。在第2-10章的开头，没有提到任何应用程序。我也没有在其他地方找到它们。

一本关于随机循环量的书将成为经典（就像戴维斯的著作是关于非随机循环量的书一样），无论该书的优缺点如何，都尚未编写。这样的书将包含应用程序，还将提供全面的调查，将随机循环类型矩阵的理论与模式化随机矩阵的更一般理论，甚至更广泛地与随机矩阵的理论联系起来。