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.

/pdf/random-graph-dynamics-5durp7r63x.pdf

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.

/pdf/random-graphs-and-complex-networks-2ht4nfa9l0.pdf

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

We consider a renewal-like recursion and prove that the solution is polynomially decaying asymptotically under suitable conditions. We prove similar results for the corresponding integral equation. In both cases coefficients and functions are of more general form than in the classic cases.

Asymptotics of a renewal-like recursion and an integral equation

Arithmetics of aging distributions: Maximum

All multivariate random variables with finite variances are univariate functions of uncorrelated random variables and if the multivariate distribution is absolutely continuous then these univariate functions are piecewise linear. They can be independent of the correlations in the Gaussian case.

Representations by uncorrelated random variables

Random Walks on de Bruijn Graphs.

http://link.springer.com/content/pdf/10.1007/978-94-009-5438-0_15.pdf

Tamás F. Móri

Papers

Asymptotics of a renewal-like recursion and an integral equation

Arithmetics of aging distributions: Maximum

Representations by uncorrelated random variables

Random Walks on de Bruijn Graphs.

The Secretary Problem with Hesitating Candidates