A Graduate Course in Probability

This book grew out of the notes for a one-semester basic graduate course in probability. As the title suggests, it is meant to be an introduction to probability and could serve as textbook for a year long text for a basic graduate course. It assumes some familiarity with measure theory and integration so in this book we emphasize only those aspects of measure theory that have special probabilistic uses.The book covers the topics that are part of the culture of an aspiring probabilist and it is guided by the author's personal belief that probability was and is a theory driven by examples. The examples form the main attraction of this subject. For this reason, a large book is devoted to an eclectic collection of examples, from classical to modern, from mainstream to 'exotic'. The text is complemented by nearly 200 exercises, quite a few nontrivial, but all meant to enhance comprehension and enlarge the reader's horizons.While teaching probability both at undergraduate and graduate level the author discovered the revealing power of simulations. For this reason, the book contains a veiled invitation to the reader to familiarize with the programing language R. In the appendix, there are a few of the most frequently used operations and the text is sprinkled with (less than optimal) R codes. Nowadays one can do on a laptop simulations and computations we could only dream as an undergraduate in the past. This is a book written by a probability outsider. That brings along a bit of freshness together with certain 'naiveties'.Contents:Foundations:Measurable SpacesMeasures and IntegrationInvariants of Random VariablesConditional ExpectationWhat are Stochastic Processes?ExercisesLimit TheoremsThe Law of Large NumbersThe Central Limit TheoremConcentration InequalitiesUniform Laws of Large NumbersThe Brownian motionExercisesMartingalesBasic Facts about MartingalesLimit Theorems: Discrete TimeContinuous Time MartingalesExercisesMarkov ChainsMarkov ChainsThe dynamics of homogeneous Markov chainsAsymptotic behaviorElectric networksFinite Markov chainsExercisesElements of Ergodic TheoryThe Ergodic TheoremApplicationsExercisesAppendix A: A Few Useful factsThe Gamma functionBasic Invariants of Frequently Used Probability DistributionsA Glimpse at RReadership: Graduate students, advanced undergraduates, mathematicians. Suitable for a first year graduate course in probability. Read more