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

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).

Probability and Measure

This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

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Random Graphs and Complex Networks

The study of random graphs has become very popular for real-life network modeling, such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice Zd, d ≥ 1, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model, various geometric properties, such as the percolation behavior, the degree distribution and graph distances, have been analyzed. In the present paper, we complement the picture of graph distances and we prove continuity of the percolation probability in the phase transition point. We also provide an illustration of the model connected to financial networks.

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Inhomogeneous Long-Range Percolation for Real-Life Network Modeling

We first discuss the convergence in probability and in distribution of the spectral norm of scaled Toeplitz, circulant, reverse circulant, symmetric circulant, and a class of k-circulant matrices when the input sequence is independent and identically distributed with finite moments of suitable order and the dimension of the matrix tends to ∞. When the input sequence is a stationary two-sided moving average process of infinite order, it is difficult to derive the limiting distribution of the spectral norm, but if the eigenvalues are scaled by the spectral density, then the limits of the maximum of modulus of these scaled eigenvalues can be derived in most of the cases.

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Spectral Norm of Circulant-Type Matrices

We present a unified approach to limiting spectral distribution (LSD) of patterned matrices via the moment method. We demonstrate relatively short proofs for the LSD of common matrices and provide insight into the nature of different LSD and their interrelations. The method is flexible enough to be applicable to matrices with appropriate dependent entries, banded matrices, and matrices of the form Ap = 1 XX 0 where X is a p × n matrix with real entries and p ! 1 with n = n(p) ! 1 and p/n ! y with 0 � y < 1. This approach raises interesting questions about the class of patterns for which LSD exists and the nature of the possible limits. In many cases the LSD are not known in any explicit forms and so deriving probabilistic properties of the limit are also interesting issues.

Patterned random matrices and method of moments

Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues in a completely different way. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.

From random matrices to long range dependence

Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process. This is similar to the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues after a different scaling. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.

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Rajat Subhra Hazra

Papers

Inhomogeneous Long-Range Percolation for Real-Life Network Modeling

Spectral Norm of Circulant-Type Matrices

Patterned random matrices and method of moments

From random matrices to long range dependence

From random matrices to long range dependence