Titlebook: Random Walk, Brownian Motion, and Martingales; Rabi Bhattacharya,Edward C. Waymire Textbook 2021 Springer Nature Switzerland AG 2021 Stoch

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ArcSine Law Asymptotics,y. The implicit symmetry of this scenario results in the counterintuitive phenomena that in a long series of plays it is not unlikely that one of the players will remain on the winning side while the other player loses for more than half of the series. This chapter derives the distribution of (a) th

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Textbook 2021n motion, and martingales. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study...Consisting of many

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The Simple Random Walk II: First Passage Times,tric random walk, referred to as null recurrence, showing that while the walker is certain to reach ., the expected time .. This refinement involves an application of Stirling’s asymptotic formula for .!, for which a proof is also provided. An extension to random walks on the integers that do not skip integer states to the left is also given.

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The Functional Central Limit Theorem (FCLT),otion, regardless of the specific random walk increments, with a finite second moment. The proof given here is by a beautiful technique of Skorokhod in which the random walk paths are embedded within the Brownian motion.

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The Poisson Process, Compound Poisson Process, and Poisson Random Field,s a fundamentally important example from the perspective of both applications and general representations of processes with independent increments. As such it may be viewed as a continuous parameter generalization of the random walk.

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The Simple Random Walk I: Associated Boundary Value Distributions, Transience, and Recurrence,ce, are identified in the course of the analysis. Recurrence is a form of “stochastic periodicity” in which the process revisits a state (or arbitrarily small neighborhood) infinitely often, while transience refers to the phenomena in which there are at most finitely many returns.