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

Convergence of Probability Measures

Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

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

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Stochastic Interacting Systems: Contact, Voter and Exclusion Processes

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

We prove a weak limit theorem which relates the large time behavior of the maximum of a branching Brownian motion to the limiting value of a certain associated martingale. This exhibits the minimal velocity travelling wave for the KPP-Fisher equation as a translation mixture of extreme-value distributions. We also show that every particle in a branching Brownian motion has a descendant at the frontier at some time. A final section states several conjectures concerning a hypothesized stationary "standing wave of particles" process and the relationship of this process to branching Brownian motion.

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A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion

0. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Renewal theorems in symbolic dynamics . . . . . . . . . . . . 1. Background: Shifts, suspension flows, thermodynamic formalism . 2. 3. 4. 5. 6. 7. 1 5 5 Renewal measures and renewal theorems . . . . . . . . . . . . . . 7 A modification for finite sequences . . . . . . . . . . . . . . . . . 10 Equidistribution theorems . . . . . . . . . . . . . . . . . . . . . . 13 Periodic orbits of suspension flows . . . . . . . . . . . . . . . . . 17 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . 20 Perturbation theory for Perron-Frobenius operators . . . . . . . . 23 8. Fourier analysis of the renewal equation . . . . . . . . . . . . . . 26 Part II. Applications to discrete groups . . . . . . . . . . . . . . . . . 32 9. Symbolic dynamics for Schottky groups . . . . . . . . . . . . . . 32 10. Symbolic dynamics for Fuchsian groups . . . . . . . . . . . . . . 34 11. The geodesic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 12. Distribution of noneuclidean lattice points and fundamental polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 13. Packing and covering functions of the limit set . . . . . . . . . . . 43 14. Random walk and Hausdorff measure . . . . . . . . . . . . . . . . 53 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits

Hausdorff and box dimensions of certain self-affine fractals

Local limit theorems and saddlepoint approximations are given for random walks on a free group whose step distributions have finite support. The techniques used to prove these results are necessarily different from those used for random walks in Euclidean spaces, because Fourier analysis is not available; the basic tools are the elementary theory of algebraic functions and the Perron-Frobenius theory of nonnegative matrices. An application to the structure of the boundary process is also given.

/pdf/finite-range-random-walk-on-free-groups-and-homogeneous-48h5xp8xov.pdf

Finite Range Random Walk on Free Groups and Homogeneous Trees

the maximum of a branching Brownian motion to the limiting value of a certain associated martingale. This exhibits the minimal velocity travelling wave for the KPP-Fisher equation as a translation mixture of extreme-value distributions. We also show that every particle in a branching Brownian motion has a descendant at the frontier at some time. A final section states several conjectures concerning a hypothesized stationary "standing wave of particles" process and the relationship of this process to branching Brownian motion.

Steven P. Lalley

Papers

A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion

Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits

Hausdorff and box dimensions of certain self-affine fractals

Finite Range Random Walk on Free Groups and Homogeneous Trees

A conditional limit theorem for the frontier of a