Statistics for Spatial Data.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.

A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate

Introduction Estimation Some Inference Problems Residual Empirical Processes Regression Models Nonparametric Regression with Heteroscedastic Errors Model Checking under Long Memory Long Memory under Infinite Variance.

Large Sample Inference for Long Memory Processes

Conditions for the CLT for non-linear functionals of stationary Gaussian sequences are discussed, with special references to the borderline between the CLT and the non-CLT. Examples of the non-CLT for such functionals with the norming factor
 
$$\sqrt N $$

 are given.

CLT and other limit theorems for functionals of Gaussian processes

ZONES OF ATTRACTION OF SELF-SIMILAR MULTIPLE INTEGRALS D. Surgailis UDC 519.21 The goal of this paper is the proof of the theorem announced in [20]. Here we consider the "multidimensional case," i.e., convergence to self-similar fields. We give a short survey of the separate sections. In Sec. 1 the concepts needed are formulated as well as the basic result of the paper (Theorem i). The connection of this theorem with the result of Dobrushin-Major [8] is discussed as well as some similar questions. In Sec. 2 the term making the basic contribution to the distribution of the sums considered is isolated. Here we explain the idea of the following proof, which is broken up into several lemmas, and their formulations are given. The proofs of these lemmas (except for Lemma 5) are carried out to section 3. In section 4 there is proved a lemma (Lemma 1 of [20]) on the convergence with respect to distribution of "discrete multiple integrals" to "continuous" integrals of Ito-- Wiener. With the help of this lemma, the remaining Lemma 5 is proved. i. Notation for what follows: R d is d-dimensional Euclidean space, x.y, ]xl are respectively the scalar product and norm in R d, Z d is the integer-valued d-dimensional lattice. We shall write

Zones of attraction of self-similar multiple integrals

The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the “Gaussian scenario”), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with “high” amplitudes) which reveals an intermittent large time behaviour, with the velocity$\vec v(t,x)$ determined by the position of the largest initial fluctuation (discounted by the heat kernelg(t,x·)) in a neighborhood ofx. The asymptoties of such local maximum ast→∞ can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process oft-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit ast→∞.

Donatas Surgailis

Papers

A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate

Large Sample Inference for Long Memory Processes

CLT and other limit theorems for functionals of Gaussian processes

Zones of attraction of self-similar multiple integrals

Stratified structure of the Universe and Burgers' equation : a probabilistic approach