This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a… Expand

The Erdos-Renyi random graphs model, a version of the CHKNS model, helps clarify the role of randomness in the distribution of values in the discrete-time world.Expand

SummaryLet W1,..., WN be N nonnegative random variables and let
$$\mathfrak{M}$$
be the class of all probability measures on [0, ∞). Define a transformation T on
$$\mathfrak{M}$$
by letting Tμ be… Expand

Abstract We consider and compare four approaches to modeling the dynamics of spatially distributed systems: mean field approaches (described by ordinary differential equations) in which every… Expand

The results suggest that the different length distributions among organisms and repeat motifs can be explained by a simple difference in slippage rates and that selective constraints on length need not be imposed.Expand

The simplest growth models. The voter model. The biased voter model. The contact process. One-dimensional discrete time models. Percolation in two dimensions. Mandelbrot's percolation process.… Expand