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Probability: Theory and Examples
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
Random graph dynamics
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.
Stochastic Calculus: A Practical Introduction
CHAPTER 1. BROWNIAN MOTION Definition and Construction Markov Property, Blumenthal's 0-1 Law Stopping Times, Strong Markov Property First Formulas CHAPTER 2. STOCHASTIC INTEGRATION Integrands:
Fixed points of the smoothing transformation
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
The Importance of Being Discrete (and Spatial)
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
Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations.
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.
Lecture notes on particle systems and percolation
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.
Brownian motion and martingales in analysis
Brownian motion. Stochastic integration. Conditioned Brownian motions. Boundary limits of harmonic functions. Complex Brownian motion and analytic functions. Hardy spaces and related spaces of