6.262.Lec7

6.262.Lec7 - DISCRETE STOCHASTIC PROCESSES Lecture 7 Review...

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6.262 Lecture 7 2/24/2010 1 DISCRETE STOCHASTIC PROCESSES Lecture 7 Review - Countable additivity axiom Continuity - like property of probability Borel - Cantelli Lemma Strong Law of Large Numbers Statement and Proof of Weaker Version #1 of Strong Law Statement and Proof of Weaker Version #2 of Strong Law Finite State Markov Chains Classification of States Matrix Approach
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6.262 Lecture 7 2/24/2010 2 1, 2, A A −−− 1 . n n B B = = U 123 BBB ⊆⊆⊆ 12 ,, BB CC 1 . n n = = I Axiom of Probability: Countable Additivity Probability is countably additive for disjoint sets. More specifically, if is a countable collection of disjoint sets, then 11 1 {} { } lim () n nk k n kk n PA P A P A →∞ == = ∑∑ U The following lemma follows from this axiom. Lemma 1 – Probability has a Property Reminescent of Continuity Let be an increasing sequence of sets, so that , and define Then . lim n n PB →∞ = (1) Similarly, let be a decreasing sequence of sets, so that , CCC ⊇−−− and define Then {} l im{ } . n n P CP C →∞ = (2)
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6.262 Lecture 7 2/24/2010 3 12 ,, EE −−− We used the continuity property to prove the following interesting result. Lemma 2: Borel-Cantelli Lemma (“Zero-One Law”) A) Let be a sequence of events. If 1 {} k k PE = < ∞ then P{an infinite number of events occur}=0. B) If are independent and 1 , k k = = ∞ then P{an infinite number of events occur}=1.
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6.262 Lecture 7 2/24/2010 4 Strong Law of Large Numbers Theorem 1 Averages of iid random variables converge almost surely to the mean. More precisely: Let X 1 , X 2 , - - - be a sequence of iid random variables with finite expectation (i.e.,E[|X|] < ) and let S n= X 1 +X 2 +---+X n , n Then i.e., 1. () ([ ] ) 1 | n n S PE X n ω →∞ →= .. [] , as n n S EX n →∞
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6.262 Lecture 7 2/24/2010 5 The strong law is very strong as stated, because the assumptions are so weak (only the mean need exist). The proof under these assumptions is beyond the range of this course. We will, however, prove the following three weaker versions of the strong law under the following stronger sets of assumptions: Let X 1 , X 2 , - - - be a sequence of iid random variables, with S n= X 1 +X 2 +---+X n : Version 2: (Next. See Gallager’s notes, sect.1.5.5) If Xn has a moment generating function g X (r) that exists in an open interval containing 0 (and therefore moments of all orders exist), then Version 3: (Next. Uses Borel – Cantelli Lemma) If Xn has at least a fourth moment, then Version 4: (Near end of course. Uses Martingale properties.) If Xn has at least a second moment, then .. [] . as n n S E X n →∞ 4 (i.e., [ ] < ) n EX . n n S E X n →∞ 2 (i.e., [ ] < , equivalently, < ) X σ . n n S E X n →∞
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6.262 Lecture 7 2/24/2010 6 1 {} k k PE = < ∞ 12 ,, EE −−− () { lim [ ]} = 1, i.e., > 0, { [ ]| > infinitely often} = 0 || nn n SS X P E X ω ωε ε →∞ =∀ 1 < .
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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.Lec7 - DISCRETE STOCHASTIC PROCESSES Lecture 7 Review...

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