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hw5sol - Stat 310A/Math 230A Theory of Probability Homework...

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Stat 310A/Math 230A Theory of Probability Homework 5 Solutions Andrea Montanari Due on November 4, 2010 Exercises on the law of large numbers and Borel-Cantelli Exercise [2.1.5] Let > 0 and pick K = K ( ) finite such that if k K then r ( k ) . Applying the Cauchy-Schwarz inequality for X i - E X i and X j - E X j we have that Cov( X i , X j ) [Var( X i )Var( X j )] 1 / 2 r (0) < for all i, j . Thus, breaking the double sum in Var( S n ) = n i,j =1 Cov( X i , X j ) into { ( i, j ) : | i - j | < K } and { ( i, j ) : | i - j | ≥ K } gives the bound Var( S n ) 2 Knr (0) + n 2 . Dividing by n 2 we see that lim sup n Var( n - 1 S n ) . Since > 0 is arbitrary and E S n = n x , we have that n - 1 S n L 2 x (with convergence in probability as well). Exercise [2.1.13] We have E | X 1 | = k =2 1 / ( ck log k ) = . On the other hand, for n N n P ( | X 1 | ≥ n ) = n c X k = n 1 k 2 log k n c Z n - 1 1 x 2 log x d x = n c Z log( n - 1) e - z z d z n c log( n - 1) Z log( n - 1) e - z d z = n c ( n - 1) log( n - 1) .
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