Random Walk: A Modern Introduction
Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
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Contents
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A MODERN INTRODUCTION 1 Introduction
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1 |
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A MODERN INTRODUCTION 2 Local central limit theorem
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21 |
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A MODERN INTRODUCTION 3 Approximation by Brownian motion
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72 |
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A MODERN INTRODUCTION 4 The Greens function
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87 |
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A MODERN INTRODUCTION 5 Onedimensional walks
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123 |
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A MODERN INTRODUCTION 6 Potential theory
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144 |
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A MODERN INTRODUCTION 7 Dyadic coupling
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205 |
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A MODERN INTRODUCTION 8 Additional topics on simple random walk
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225 |
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A MODERN INTRODUCTION 9 Loop measures
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247 |
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A MODERN INTRODUCTION 10 Intersection probabilities for random walks
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297 |
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A MODERN INTRODUCTION 11 Looperased random walk
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307 |
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A MODERN INTRODUCTION Appendix
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326 |
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A MODERN INTRODUCTION Bibliography
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360 |
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Common terms and phrases
AC Zd aperiodic asymptotics B₁ bounded bounded function Brownian motion c₁ cap(A characteristic function consider constant continuous-time convergence Corollary covariance matrix define denote eigenvalues error term Exercise exists expected number finite follows GA(x gambler's ruin gives graph Green's function harmonic Harnack inequality Hence implies increment distribution independent inequality intersect irreducible lattice LCLT Lemma LERW logn loop-erased Markov chain Markov property martingale mean zero min{j Note number of visits one-dimensional optional sampling theorem P{Sn particular Pn(x Poisson kernel positive integer potential kernel probability measure probability space Proposition random variables random walk starting result satisfying Section self-avoiding path Show simple random walk spanning trees sufficiently large Suppose symmetric third moments transient transition probability walk with increment write