6.262 Lect. 6 2/19/2010Discrete Stochastic Processes1DISCRETESTOCHASTICPROCESSES Lecture 6 Convergence of Sequences of Random Variables Review: Sequences of random variables: four examplesFour progressively weaker types of convergence(Review): Sure convergence of a sequence of random variables.Almost sure convergence and the strong law of large numbersConvergence 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
has intentionally blurred sections.
Sign up to view the full version.
6.262 Lect. 6 2/19/2010Discrete Stochastic Processes2Math Notation ∀means “for each” (equivalently, “for every” or “for all”). ∃means “there exists”. AB⇒means that “whenever Ais true, Bis also true.” s.t. means “such that”. nXX→means “nXconverges to X as n→∞,” though we also need to specify what sort of objects nXand Xare (numbers, functions, random variables) and what kind of convergence we mean.