Stochastic limit theory. An introduction for econometricians.
Repr.

*(English)*Zbl 0904.60002
Advanced Texts in Econometrics. Oxford: Oxford Univ. Press. xxii, 539 p. (1997).

This book is about limit theorems in probability theory in the context of econometrics. It is written for econometricians and therefore includes a careful and thorough introductory part covering the required mathematical background, but most of the material is also useful to mathematicians with an interest in the area of stochastic limit theorems for sequences of dependent random variables. Given the wealth of carefully presented material covered on more than 500 pages, the price of £25 for (the paperback version of) the book is a very pleasant surprise.

In more detail, the book contains 30 chapters grouped into six roughly equal parts. Part I covers general mathematical background material and in particular measure theory and integration, metric spaces and topology. Part II about probability contains chapters about probability spaces, random variables, expectations, conditioning and characteristic functions. Part III then deals with stochastic processes in discrete time and more precisely with those topics that are of importance for econometric modelling and asymptotics. The subjects include dependence, mixing, martingales, mixingales and near-epoch dependence. Broadly speaking, the overall goal here is to present in detail various forms of dependence structures for sequences of random variables that still allow one to obtain limit theorems.

In part IV, stochastic limit theory then starts in earnest. After an explanation of various convergence concepts from probability, weak and strong laws of large numbers are discussed in detail for a wide range of situations. A chapter on uniform stochastic convergence concludes this part. Part V is concerned with the central limit theorem. It explains weak convergence of distributions on the real line, presents first the classical CLT and then CLTs for dependent processes including martingales and stationary ergodic sequences, and concludes with some extensions to estimated normalizations, random norming, error estimates and the multivariate CLT. Finally, part VI is about functional limit theorems. It first discusses weak convergence in general metric spaces and then on the space of continuous functions and on the Skorokhod space. Chapter 29 presents several results about weak convergence to (possibly time-changed) Brownian motion, and the final Chapter 30 contains more recent results on weak convergence to stochastic integrals of a Brownian motion.

In summary, this book is an excellent self-contained presentation of weak convergence results for a very broad range of dependent sequences of random variables having finite variances. As noted in the introduction, one notable missing topic is convergence to stable laws for sequences with infinite variances. In view of the amount of recent work on financial time series, it might be useful to include such results in a next edition of an otherwise very comprehensive book.

In more detail, the book contains 30 chapters grouped into six roughly equal parts. Part I covers general mathematical background material and in particular measure theory and integration, metric spaces and topology. Part II about probability contains chapters about probability spaces, random variables, expectations, conditioning and characteristic functions. Part III then deals with stochastic processes in discrete time and more precisely with those topics that are of importance for econometric modelling and asymptotics. The subjects include dependence, mixing, martingales, mixingales and near-epoch dependence. Broadly speaking, the overall goal here is to present in detail various forms of dependence structures for sequences of random variables that still allow one to obtain limit theorems.

In part IV, stochastic limit theory then starts in earnest. After an explanation of various convergence concepts from probability, weak and strong laws of large numbers are discussed in detail for a wide range of situations. A chapter on uniform stochastic convergence concludes this part. Part V is concerned with the central limit theorem. It explains weak convergence of distributions on the real line, presents first the classical CLT and then CLTs for dependent processes including martingales and stationary ergodic sequences, and concludes with some extensions to estimated normalizations, random norming, error estimates and the multivariate CLT. Finally, part VI is about functional limit theorems. It first discusses weak convergence in general metric spaces and then on the space of continuous functions and on the Skorokhod space. Chapter 29 presents several results about weak convergence to (possibly time-changed) Brownian motion, and the final Chapter 30 contains more recent results on weak convergence to stochastic integrals of a Brownian motion.

In summary, this book is an excellent self-contained presentation of weak convergence results for a very broad range of dependent sequences of random variables having finite variances. As noted in the introduction, one notable missing topic is convergence to stable laws for sequences with infinite variances. In view of the amount of recent work on financial time series, it might be useful to include such results in a next edition of an otherwise very comprehensive book.

Reviewer: M.Schweizer (Berlin)