## Random Walk: A Modern IntroductionRandom 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

1 | |

A MODERN INTRODUCTION 2 Local central limit theorem | 21 |

A MODERN INTRODUCTION 3 Approximation by Brownian motion | 72 |

A MODERN INTRODUCTION 4 The Greens function | 87 |

A MODERN INTRODUCTION 5 Onedimensional walks | 123 |

A MODERN INTRODUCTION 6 Potential theory | 144 |

A MODERN INTRODUCTION 7 Dyadic coupling | 205 |

A MODERN INTRODUCTION 8 Additional topics on simple random walk | 225 |

A MODERN INTRODUCTION 9 Loop measures | 247 |

A MODERN INTRODUCTION 10 Intersection probabilities for random walks | 297 |

A MODERN INTRODUCTION 11 Looperased random walk | 307 |

A MODERN INTRODUCTION Appendix | 326 |

A MODERN INTRODUCTION Bibliography | 360 |

361 | |

363 | |

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### Common terms and phrases

aperiodic assume asymptotics bounded bounded function Brownian motion C2 log characteristic function consider constant continuous-time convergence Corollary covariance matrix define denote eigenvalues error term Exercise exists expected number finite ﬁrst gambler's ruin gives graph Green's function Green’s Harnack inequality Hence implies increment distribution independent inequality intersect irreducible lattice LCLT Lemma LERW logn loop-erased loop-erased random walk Markov chain Markov property martingale mean zero Note number of visits optional sampling theorem p e V'd particular pn(x Poisson kernel positive integer potential kernel probability measure probability space Proof Let Proposition random variables random walk starting result satisfying Section self-avoiding path Show simple random walk spanning trees strong Markov property subset sufficiently large Suppose that p e symmetric third moments transient transition probability walk in Zd walk with increment write x e Zd