## Probability TheoryThis volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation. In the first part of the book, characteristic functions are introduced, followed by the study of weak convergence of probability distributions. Then both the weak and strong limit theorems for sums of independent random variables are proved, including the weak and strong laws of large numbers, central limit theorems, laws of the iterated logarithm, and the Kolmogorov three series theorem. The first part concludes with infinitely divisible distributions and limit theorems for sums of uniformly infinitesimal independent random variables. The second part of the book mainly deals with dependent random variables, particularly martingales and Markov chains. Topics include standard results regarding discrete parameter martingales and Doob's inequalities. The standard topics in Markov chains are treated, i.e., transience, and null and positive recurrence. A varied collection of examples is given to demonstrate the connection between martingales and Markov chains. Additional topics covered in the book include stationary Gaussian processes, ergodic theorems, dynamic programming, optimal stopping, and filtering. A large number of examples and exercises is included. The book is a suitable text for a first-year graduate course in probability. |

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### Contents

Weak Convergence | 19 |

Independent Sums | 35 |

Dependent Random Variables | 73 |

Martingales | 109 |

14 | 119 |

19 | 127 |

Stationary Stochastic Processes | 131 |

22 | 144 |

Dynamic Programming and Filtering | 157 |

163 | |

165 | |

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apply assume bounded calculate called characteristic function choice Clearly complete conclude conditional Consider constant construct continuous convergence corresponding countably additive deﬁne defined DEFINITION denote density depends determined difference distribution equal equation equivalent ergodic establish estimate example EXERCISE exists fact ﬁnite function f Gaussian given implies important independent induction inequality integrable interval invariant measure lemma lim sup limit linear Markov chain Markov process martingale mean measurable function nonnegative particular positive possible probability distribution probability measure problem PROOF prove question random variables recurrent REMARK representation respect satisfying sequence simple space starting stationary step stopping subsets Suppose sure theorem ticket transition probability true uniform uniformly unique variance