Even though the title of the book asserts that the book contains applications, in the first lines of the preface the author disclaims that perhaps the sub-title should have been: “Applicable to econometrics.” As the author says, it is true that each chapter aims at self-containment. All basic material for each chapter is either developed or recalled in the first sections of the chapter, and the proofs of most of the statements are included. Each important class of results is grouped in sections, with the sub-section titles explaining what is being done. Even though the details of most proofs are explicit, whether “the book can be read without paper and pencil” I don’t know. Misprints are rare, and if Ph.J. Davis’ book, “The thread: a mathematical yarn.” Boston etc.: Birkhäuser (1983; Zbl 0514.00005), ever gets rewritten, a new spelling (transliteration) of a famous name (Chebyshev) should be added to his list.
Loosely speaking, this books deals with two kinds of problems. One of them is the convergence of series of the type \(\sum_{j\geq 1} \psi_{n-j} e_{j}\) which arise when examining solutions of the equation \(Y_n = \psi_n Y_{n-1} + e_n\). The generic case treated is when \(\{e_n \}\) is a martingale difference process. The other is, in its simplest version, that of estimating \(X\) in \(Y = BX + e\). Here \(Y \in{\mathbb R}^n\) is a data vector, and \(e\) is again a martingale difference (vector). Here it is the size of the data vector that increases as the size of the design (or transfer) matrix \(B\) increases. The interest here lies the standard statistical properties of the estimator \(\hat X\). In chapter one, a collection of very basic concepts and notation needed for the whole book is established. Chapter two is devoted to many aspects of approximation theory in \(\mathcal L_p\) spaces.
In chapter three the main subject of the book is taken up: the convergence of linear and quadratic forms. In particular the examination of results of the type of the central limit theorem, that is of weighted sums of random variables (martingale differences) is commenced. The interest in the results is that the link between regressors and the asymptotic properties is better established. Chapter four is devoted to the study of the asymptotic properties of linear regression coefficients in terms of the the asymptotic properties of the design matrix. In chapter five, spatial models are considered. Here the notion of auto-regression is extended to include models in which the value of the random variable at a particular point of time or space depends linearly of the values of the random variables at any other (future or past) times or arbitrary points in space. The systems can be thought of as discretized versions of integral equations with an error term. The chapter ends with a description of some open problems.
In chapter six an extension of the classical linear estimation and regression theories is carried out. By classical I mean models in which the innovation (error) terms are independent with controlled variance growth. Here the author presents a detailed description of an extension of the classical theory due to Lai and Wei. For the extension the error (innovation) terms are replaced by martingale increments, and the growth condition on their variances is replaced by the so called (extended) Marcinkiewicz-Zygmund (or Lai-Wei) conditions. Besides describing several generic martingale convergence results, he describes de asymptotic properties of the estimators and the convergence of the regression series in this case. A reference that he missed is the set of the terse lecture notes of C.-Z. Wei, [“Martingale limit theory and stochastic regression theory.” Downloadable from the web, e.g., http://www.free-ebooks.gr/eng/ebook/66]‘. Chapter seven is devoted to nonlinear estimation models. The properties of nonlinear least squares and maximum likelihood estimators are developed for the simple nonlinear regression models and for the logit models. The last chapter is a collection of basic \(L_p\)-approximable matrix valued functions, with references to possible applications.