6.262.Lec6

# 6.262.Lec6 - DISCRETE STOCHASTIC PROCESSES Lecture 6...

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6.262 Lect. 6 2/19/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 6 Convergence of Sequences of Random Variables Review: Sequences of random variables: four examples Four progressively weaker types of convergence (Review): Sure convergence of a sequence of random variables. Almost sure convergence and the strong law of large numbers Convergence in probability and the weak law of large numbers. (Review): Convergence in distribution and the central limit theorem. (Review): Convergence in mean-square The Zero - One Laws of Borel and Cantelli Axiom: Probability is Countably Additive Lemma: Probability has a Continuity-Like Property Application: Continuity of CDF from Right but not from Left Statement of Zero-One Law of Borel and Cantelli Proof of Borel-Cantelli Lemma Examples and Applications

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6.262 Lect. 6 2/19/2010 Discrete Stochastic Processes 2 Math Notation means “for each” (equivalently, “for every” or “for all”). means “there exists”. A B means that “whenever A is true, B is also true.” s.t. means “such that”. n X X means “ n X converges to X as n →∞ ,” though we also need to specify what sort of objects n X and X are (numbers, functions, random variables) and what kind of convergence we mean.