18.175: Lecture 10 Zero-one Laws And Maximal Inequalities

18.175: Lecture 10Zero-one laws and maximal inequalitiesScott SheffieldMIT18.175 Lecture 101

OutlineRecollectionsKolmogorov zero-one law and three-series theorem18.175 Lecture 102

OutlineRecollectionsKolmogorov zero-one law and three-series theorem18.175 Lecture 103

Recall Borel-Cantelli lemmasS First Borel-Cantelli lemma: IfP(An i.o.) 0. SecondBorel-Cantelli lemma: If An are independent, thenS P(An ) implies P(An i.o.) 1.n 118.175 Lecture 104n 1 P(An ) then

Recall strong law of large numbers ITheorem (strong law): If X1 , X2 , . . . are i.i.d. real-valuedSrandom variables with expectation m and An : n 1 ni 1 Xiare the empirical means then limn An m almost surely.18.175 Lecture 105

OutlineRecollectionsKolmogorov zero-one law and three-series theorem18.175 Lecture 106

OutlineRecollectionsKolmogorov zero-one law and three-series theorem18.175 Lecture 107

Kolmogorov zero-one law IConsider sequence of random variables Xn on some probabilityspace. Write Fn σ(Xn , Xn1 , . . .) and T n Fn . IT is called the tail σ-algebra. It contains the information youcan observe by looking only at stuff arbitrarily far into thefuture. Intuitively, membership in tail event doesn’t changewhen finitely many Xn are changed. IEvent that Xn converge to a limit is example of a tail event.Other examples? ITheorem: If X1 , X2 , . . . are independent and A T thenP(A) {0, 1}.18.175 Lecture 108

Kolmogorov zero-one law proof idea ITheorem: If X1 , X2 , . . . are independent and A T thenP(A) {0, 1}. IMain idea of proof: Statement is equivalent to saying that Ais independent of itself, i.e., P(A) P(A A) P(A)2 . Howdo we prove that? IRecall theorem that if Ai are independent π-systems, thenσAi are independent. IDeduce that σ(X1 , X2 , . . . , Xn ) and σ(Xn 1 , Xn 1 , . . .) areindependent. Then deduce that σ(X1 , X2 , . . .) and T areindependent, using fact that k σ(X1 , . . . , Xk ) and T areπ-systems.18.175 Lecture 109

MIT OpenCourseWarehttp://ocw.mit.edu18.175 Theory of ProbabilitySpring 2014For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

MIT OpenCourseWarehttp://ocw.mit.edu18.175 Theory of ProbabilitySpring 2014For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

Kolmogorov zero-one law. Consider sequence of random variables X. n. on some probability space. Write F σ(X. n, X. n. n. 1,.) and T . n. F. 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.