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

Convergence of Probability Measures

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

1. Overview 2. Erdos-Renyi random graphs 3. Fixed degree distributions 4. Power laws 5. Small worlds 6. Random walks 7. CHKNS model.

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Random graph dynamics

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

Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Fixed Point Theory

In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.

On random trees

In a one-parameter model for evolution of random trees, which also includes the Barabasi–Albert random graph [1], the law of large numbers and the central limit theorem are proved for the maximal degree. In the proofs martingale methods are applied.

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The Maximum Degree of the Barabási–Albert Random Tree

On relationships between the Pearson and the distance correlation coefficients

Let X be a d-dimensional standardized random variable $({\bf E}(X) = 0, \operatorname{cov} (X) = 1)$. Then for a multivariate analogue of skewness $s = {\bf E}(\| X \|^2 X)$ and kurtosis $k = {\bf E}XX^T XX^T - (d + 2)I$ we show that $\| s \|^2 \leqq {\text{tr}}\, k + 2d$. For infinitely divisible distributions $\| s \|^2 \leqq {\text{tr }}k$.

On Multivariate Skewness and Kurtosis

Let A 1 , A 2 , · ··, A n be events on a probability space. Denote by S k the k th binomial moment of the number M n of those A 's which occur. Sharp lower and upper bounds of S m will be given in terms of S k and S l . The same method can be applied for proving Bonferroni–Galambos-type inequalities.

Tamás F. Móri

Papers

On random trees

The Maximum Degree of the Barabási–Albert Random Tree

On relationships between the Pearson and the distance correlation coefficients

On Multivariate Skewness and Kurtosis

A note on the background of several Bonferroni−Galambos-type inequalities