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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

Cognitive radio (CR) is the enabling technology for supporting dynamic spectrum access: the policy that addresses the spectrum scarcity problem that is encountered in many countries. Thus, CR is widely regarded as one of the most promising technologies for future wireless communications. To make radios and wireless networks truly cognitive, however, is by no means a simple task, and it requires collaborative effort from various research communities, including communications theory, networking engineering, signal processing, game theory, software-hardware joint design, and reconfigurable antenna and radio-frequency design. In this paper, we provide a systematic overview on CR networking and communications by looking at the key functions of the physical (PHY), medium access control (MAC), and network layers involved in a CR design and how these layers are crossly related. In particular, for the PHY layer, we will address signal processing techniques for spectrum sensing, cooperative spectrum sensing, and transceiver design for cognitive spectrum access. For the MAC layer, we review sensing scheduling schemes, sensing-access tradeoff design, spectrum-aware access MAC, and CR MAC protocols. In the network layer, cognitive radio network (CRN) tomography, spectrum-aware routing, and quality-of-service (QoS) control will be addressed. Emerging CRNs that are actively developed by various standardization committees and spectrum-sharing economics will also be reviewed. Finally, we point out several open questions and challenges that are related to the CRN design.

Cognitive radio networking and communications: an overview

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Random Walk, Green Function, Equilibrium Measure.- Random Interlacements: First Definition and Basic Properties.- Random Walk on the Torus and Random Interlacements.- Poisson Point Processes.- Random Interlacements Point Process.- Percolation of the Vacant Set.- Source of Correlations and Decorrelation via Coupling.- Decoupling Inequalities.- Phase Transition of Vu.- Coupling of Point Measures of Excursions.

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An Introduction to Random Interlacements

In this paper, we provide general conditions on a one parameter family of random infinite subsets of Z d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab.17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation,” Ann Probab.24(2), 1036–1048 (1996)] for Bernoulli percolation. Finally, as a corollary, we derive new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.

On chemical distances and shape theorems in percolation models with long-range correlations

We prove a quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on $${\mathbb {Z}}^d$$
 , $$d\ge 2$$
 , with long-range correlations introduced in (Drewitz et al. in J Math Phys 55(8):083307, 2014), solving one of the open problems from there. This gives new results for random interlacements in dimension $$d\ge 3$$
 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models.

Quenched invariance principle for simple random walk on clusters in correlated percolation models

In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. We also obtain alternative proofs to the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.

We consider the interlacement Poisson point process on the space of doubly-infinite Z d -valued trajectories modulo time shift, tending to infinity at pos- itive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the random interlacement at level u of Sznitman (2010). We prove that for any u > 0, almost surely, (1) any two vertices in the random interlacement at level u are connected via at most dd/2e trajectories of the point process, and (2) there are vertices in the random interlacement at level u which can only be connected via at least dd/2e trajectories of the point process. In par- ticular, this implies the already known result of Sznitman (2010) that the random interlacement at level u is connected.

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Artem Sapozhnikov

Papers

An Introduction to Random Interlacements

On chemical distances and shape theorems in percolation models with long-range correlations

Quenched invariance principle for simple random walk on clusters in correlated percolation models

On chemical distances and shape theorems in percolation models with long-range correlations

Connectivity properties of random interlacement and intersection of random walks