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 XX 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.
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6.262 Lect. 6 2/19/2010 Discrete Stochastic Processes 3 k I Examples for Convergence of a Sequence of Random Variables Example 1: Wk is a sequence of (neither independent nor identically distributed) random variables, with W k uniformly distributed in [0, 1/k], k 1. Example 2 (“Double or Nothing”): Let , 1 k Zk be a sequence of independent binary random variables, with {} 1 02 2 kk PZ == and let n X be the product random process 1 . n nk k X Z π = = Example 3 ("Ever Rarer Wins"): Let I k be the set of integers n in the interval 1 22 n < (i.e., { } { } { } 12 3 1 , 2,3 , 4,5,6,7 , II I = etc.) From each interval I k choose one integer k N at random (i.e., N k is uniformly distributed in I k and independent of all other N j , j k). Set 1, if for some Yn N k = = 0, otherwise n Y = Then (since has 2 (k-1) elements and each positive integer 2 ln ( ) 1 n nI + ±±±±±±± , ⎣⎦ 2 ln ( ) {1 } 2 n n PY
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6.262 Lect. 6 2/19/2010 Discrete Stochastic Processes 4 Example 4 Let { } ,1 n Rn be the set of iid Bernoulli random variables with {} 1 11 . 2 rn PR =− = =+ = Define 1 1 n nk k A R n = = We will use the sequences X n , Y n , A n , B n , & C n to illustrate the various notions of convergence of a sequence of random variables. 1 n k CR = = 1 1 n k BR n = =
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6.262 Lect. 6 2/19/2010 Discrete Stochastic Processes 5 Since a random variable is a function from Ω to ± , we say a sequence of random variables ( ) n X ω converges surely to a random variable if for each 0 ε Ω the sequence of numbers ( ) 0 n X converges to the number ( ) 0 . X This is just pointwise convergence of the sequence of functions n X .
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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

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