##
**Uniform central limit theorems.**
*(English)*
Zbl 0951.60033

Cambridge Studies in Advanced Mathematics. 63. Cambridge: Cambridge University Press. xiv, 436 p. (1999).

Let \(P_n\) denote the empirical measure based on independent identically distributed random variables taking values in some measurable space \((S,{\mathcal S})\), \(n\in {\mathbf N}\), and let \(P\) denote their common probability law. A class of \(P\)-square integrable real functions \({\mathcal F}\) on \(S\) is \(P\)-Donsker if the central limit theorem holds for the sequence \(\{\nu_n(f): =\sqrt n\int fd(P_n-P)\}\) uniformly in \({\mathcal F}\), meaning that the \({\mathcal F}\)-indexed processes \(\{\nu_n(f) :f\in {\mathcal F}\}\) converge ‘in law’ to a Gaussian process \(G_P(f)\), \(f\in {\mathcal F}\), that satisfies a regularity property \((G_P\) has a version with all its sample paths bounded and uniformly continuous w.r.t. its own \(L_2\) distance). In this notation, the classical Donsker’s theorem for empirical distributions asserts that the class of indicator functions of half lines in \({\mathbf R}\) is \(P\)-Donsker for all Borel probability measures \(P\). Here, \({\mathbf R}\) is replaced by a general space, the half lines by a general class of functions, and the convergence in law, instead of taking place in \(D(0,1)\) takes place in the space of bounded functionals on \({\mathcal F}\), with the sup norm, \(\ell^\infty ({\mathcal F})\), a Banach space which is not separable (unless \({\mathcal F}\) is finite). This definition is due to several authors, notably to Dudley and to Hoffmann-Jørgensen.

The book under review is mainly about Donsker classes. There are measurablity questions because \(\nu_n: \Omega\mapsto \ell^\infty ({\mathcal F})\) is not Borel measurable in general. These questions are treated in detail in Chapters 3 and 5. The above definition includes sample path properties of Gaussian processes; these and other relevant properties of Gaussian processes are treated in Chapter 2 which, in particular, contains a complete, detailed proof of the famous Talagrand’s sample continuity theorem. When \({\mathcal F}\) is the unit ball of the dual of a separable Banach space, the \(P\)-Donsker property of \({\mathcal F}\) is equivalent to the central limit theorem holding in that space, and this has been extensively studied during the seventies (probability in Banach spaces). Aside from this ‘simpler’ case, there are two main techniques available in order to prove that a class of functions is \(P\)-Donsker that only involve properties of the class \({\mathcal F}\) with respect to \(P\) (and not w.r.t. to \(P^n\) for all \(n)\); these are the combinatorial Vapnik-Červonenkis approach (Chapters 4 and 6) and metric entropy with bracketing (Chapter 7). The Donsker property in the end reduces to maximal inequalities and these techniques involve metric entropy and chaining as in Gaussian processes to deal with them, hence, they have to do with the ‘smallness’ of the class \({\mathcal F}\). Estimating the size of classes of functions, or their metric entropy with respect to certain metrics, is an important, some times difficult, matter in approximation theory. Chapter 8 does this for important classes (classes of differentiable functions and of sets with differentiable boundaries, of convex sets, of lower layers, of convex hull in Hilbert space). The book also treats other important related subjects such as uniform (in \(P)\) and universal central limit theorems and laws of large numbers, the invariance principle, the two sample case, the bootstrap central limit theorem for empirical processes, and classes and sets or functions too large, but borderline, to be Donsker classes.

This book is very well structured, excellently written and remarkably self-contained. The material that is given without proofs is placed in doubly starred sections and is not used in the rest of the book; and sections with one star contain material that is not used in the sequel, except perhaps for other starred sections. Each chapter ends with a set of exercises and with historical notes. These qualities make the book very well suited for graduate courses, and it is also an excellent reference book. Among the subjects only mentioned, but not developed, perhaps we should mention Talagrand’s law of large numbers, the uniform (in \(P)\) central limit theorem, Gaussian and random entropy characterizations of Donsker classes, the law of the iterated logarithm, and deviation and concentration exponential inequalities. But, developing these subjects in the self-contained style and at the level of rigor and detail of this book would require another one of at least the same size.

The book under review is mainly about Donsker classes. There are measurablity questions because \(\nu_n: \Omega\mapsto \ell^\infty ({\mathcal F})\) is not Borel measurable in general. These questions are treated in detail in Chapters 3 and 5. The above definition includes sample path properties of Gaussian processes; these and other relevant properties of Gaussian processes are treated in Chapter 2 which, in particular, contains a complete, detailed proof of the famous Talagrand’s sample continuity theorem. When \({\mathcal F}\) is the unit ball of the dual of a separable Banach space, the \(P\)-Donsker property of \({\mathcal F}\) is equivalent to the central limit theorem holding in that space, and this has been extensively studied during the seventies (probability in Banach spaces). Aside from this ‘simpler’ case, there are two main techniques available in order to prove that a class of functions is \(P\)-Donsker that only involve properties of the class \({\mathcal F}\) with respect to \(P\) (and not w.r.t. to \(P^n\) for all \(n)\); these are the combinatorial Vapnik-Červonenkis approach (Chapters 4 and 6) and metric entropy with bracketing (Chapter 7). The Donsker property in the end reduces to maximal inequalities and these techniques involve metric entropy and chaining as in Gaussian processes to deal with them, hence, they have to do with the ‘smallness’ of the class \({\mathcal F}\). Estimating the size of classes of functions, or their metric entropy with respect to certain metrics, is an important, some times difficult, matter in approximation theory. Chapter 8 does this for important classes (classes of differentiable functions and of sets with differentiable boundaries, of convex sets, of lower layers, of convex hull in Hilbert space). The book also treats other important related subjects such as uniform (in \(P)\) and universal central limit theorems and laws of large numbers, the invariance principle, the two sample case, the bootstrap central limit theorem for empirical processes, and classes and sets or functions too large, but borderline, to be Donsker classes.

This book is very well structured, excellently written and remarkably self-contained. The material that is given without proofs is placed in doubly starred sections and is not used in the rest of the book; and sections with one star contain material that is not used in the sequel, except perhaps for other starred sections. Each chapter ends with a set of exercises and with historical notes. These qualities make the book very well suited for graduate courses, and it is also an excellent reference book. Among the subjects only mentioned, but not developed, perhaps we should mention Talagrand’s law of large numbers, the uniform (in \(P)\) central limit theorem, Gaussian and random entropy characterizations of Donsker classes, the law of the iterated logarithm, and deviation and concentration exponential inequalities. But, developing these subjects in the self-contained style and at the level of rigor and detail of this book would require another one of at least the same size.

Reviewer: E.Gine (Storrs)

### MSC:

60F17 | Functional limit theorems; invariance principles |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G07 | General theory of stochastic processes |

60G15 | Gaussian processes |

60G17 | Sample path properties |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

62E20 | Asymptotic distribution theory in statistics |

68Q25 | Analysis of algorithms and problem complexity |