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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 limsup 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...
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online