6441lecture4 - LECTURE 4 Convergence and Asymptotic...

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LECTURE 4 Convergence and Asymptotic Equipartition Property Last time: Fano’s Inequality Stochastic Processes Entropy Rate Hiden Markov Process Lecture outline Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Reading: Chapter 3.
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Convergence of Random Variables A sequence of maps Ω X converge, w.o.l.g., to 0. Pointwise convergence: for any ω Ω, X n ( ω ) 0. Goal The Law of Large Numbers: the av- erage of a sequence of i.i.d. r.v.s converges to the mean. n 1 lim X n E [ X ] n n →∞ i =1 Need weaker notions.
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Types of convergence Almost sure convergence (also called con- vergence with probability 1) P �� ω : lim Y n ( ω ) = Y ( ω ) �� = 1 n →∞ write Y n Y a.s. . Mean-square convergence: lim E [ Y n Y 2 ] = 0 n →∞ | | Convergence in probability: > 0 lim P ( { ω : Y n ( ω ) Y ( ω ) > } ) = 0 n →∞ | | Convergence in distribution: the cumula- tive distribution function (CDF) F n ( y ) = P r ( Y n y ) satisfy lim F n ( y ) F Y ( y ) n →∞ at all y for which F is continuous.
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Relations among types of convergence Venn diagram of relation:
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Weak Law of Large Numbers X 1 , X 2 , . . . i.i.d.
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