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

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.

/pdf/processes-on-unimodular-random-networks-5bf1dws33o.pdf

Processes on Unimodular Random Networks

Geometry of sets and measures in euclidean spaces fractals and rectifiability

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Random Walks in Random Environment.

We consider a supercritical Bernoulli percolation model in , d2, and study the simple symmetric random walk on the infinite percolation cluster. The aim of this paper is to prove the almost sure (q...

/pdf/quenched-invariance-principles-for-random-walks-on-4m8mbdx7ux.pdf

Quenched invariance principles for random walks on percolation clusters

We prove an almost sure invariance principle for a random walker among i.i.d. conductances in ℤ
 d
 , d≥2. We assume conductances are bounded from above but we do not require that they are bounded from below.

/pdf/quenched-invariance-principles-for-random-walks-with-random-l4ofea0zf3.pdf

Quenched Invariance Principles for Random Walks with Random Conductances

We prove that the heat kernel on the infinite Bernoulli percolation cluster in \Zd almost surely decays faster than t−d/2. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities. Some of the results of this paper were previously announced in the note of Mathieu and Remy [C. R. Acad. Sci. Paris Ser. I Math. 332 (2001) 927--931].

/pdf/isoperimetry-and-heat-kernel-decay-on-percolation-clusters-2hx528zsd0.pdf

Isoperimetry and heat kernel decay on percolation clusters

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.

/pdf/harmonic-measures-versus-quasiconformal-measures-for-1ivpow5xjt.pdf

Harmonic measures versus quasiconformal measures for hyperbolic groups

We prove that the heat kernel on the infinite Bernoulli percolation cluster in Z^d almost surely decays faster than t^{-d/2}. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities.

Pierre Mathieu

Papers

Quenched invariance principles for random walks on percolation clusters

Quenched Invariance Principles for Random Walks with Random Conductances

Isoperimetry and heat kernel decay on percolation clusters

Harmonic measures versus quasiconformal measures for hyperbolic groups

Isoperimetry and heat kernel decay on percolation clusters