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18.175: Lecture 10 Zero-one laws and maximal inequalities
Scott Sheffield
MIT
18.175 Lecture 10
Outline
Recollections Kolmogorov zero-one law and three-series theorem
18.175 Lecture 10
18.175: Lecture 10 Zero-one laws and maximal inequalities Scott - - PowerPoint PPT Presentation
18.175: Lecture 10 Zero-one laws and maximal inequalities Scott - - PowerPoint PPT Presentation
18.175: Lecture 10 Zero-one laws and maximal inequalities Scott Sheffield MIT 1 18.175 Lecture 10 Outline Recollections Kolmogorov zero-one law and three-series theorem 2 18.175 Lecture 10 Outline Recollections Kolmogorov zero-one law and
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Outline
Recollections Kolmogorov zero-one law and three-series theorem
18.175 Lecture 10
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Recall Borel-Cantelli lemmas
S∞
First Borel-Cantelli lemma: If
P(An) < ∞ then
n=1
P(An i.o.) = 0.
Second Borel-Cantelli lemma: If An are independent, then
S∞ P(An) = ∞ implies P(An i.o.) = 1.
n=1
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- Recall strong law of large numbers
Theorem (strong law): If X1, X2, . . . are i.i.d. real-valued S
−1 n
random variables with expectation m and An := n
i=1 Xi
are the empirical means then limn→∞ An = m almost surely.
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Outline
Recollections Kolmogorov zero-one law and three-series theorem
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Outline
Recollections Kolmogorov zero-one law and three-series theorem
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- Kolmogorov zero-one law
Consider sequence of random variables Xn on some probability
- space. Write F = σ(Xn, Xn1 , . . .) and T = ∩nF.
n n
T is called the tail σ-algebra. It contains the information you can observe by looking only at stuff arbitrarily far into the
- future. Intuitively, membership in tail event doesn’t change
when finitely many Xn are changed. Event that Xn converge to a limit is example of a tail event. Other examples? Theorem: If X1, X2, . . . are independent and A ∈ T then P(A) ∈ {0, 1}.
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- Kolmogorov zero-one law proof idea
Theorem: If X1, X2, . . . are independent and A ∈ T then P(A) ∈ {0, 1}. Main idea of proof: Statement is equivalent to saying that A is independent of itself, i.e., P(A) = P(A ∩ A) = P(A)2 . How do we prove that? Recall theorem that if Ai are independent π-systems, then σAi are independent. Deduce that σ(X1, X2, . . . , Xn) and σ(Xn+1, Xn+1, . . .) are
- independent. Then deduce that σ(X1, X2, . . .) and T are
independent, using fact that ∪k σ(X1, . . . , Xk ) and T are π-systems.
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MIT OpenCourseWare http://ocw.mit.edu
18.175 Theory of Probability
Spring 2014 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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MIT OpenCourseWare http://ocw.mit.edu
18.175 Theory of Probability
Spring 2014 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.