h6 - Z ; that is, as n with n Nr Z . 6b: Do Billingsleys...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Probability II MAP6473 4810 Homework-6 Prof. JLF King 4May2010 Reading. Please read the ergodic theory chap- ter of Billingsley. Notation in force. For sets E,A N , say that E eventually includes A if E A [ N .. ) for some sufficiently large N . 6a: Suppose that A 1 ,A 2 ,... are zero-density subsets of N . Then there exists a zero-density set E which eventually-includes each A j . [ Hint: Think Cantor diag- onalization.] Consider a sequence ~ X of non-negative reals. For each ε > 0, let N ε := { n | x n > ε } . For each ε , suppose that each N ε is a zero-density set. Prove that there exists a zero-density set Z N so that x n 0 off of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Z ; that is, as n with n Nr Z . 6b: Do Billingsleys problem, 24 . 7 P. 326. Here ( T : X, X , P ) is a mixing transformation. A fnc : X [ , ) , with R X () d P = 1, gives rise to a new probability measure (the text calls it P ) by ( A ) := Z A () d P . Prove, for each measurable B , that ( T-n ( B )) P ( B ) as n . [ Hint: You might rst want to consider the case where is a ( scaled ) indicator function [1 / P ( A )] 1 A .]...
View Full Document

Ask a homework question - tutors are online