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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2017 • Question 4

Question

Math 6010 • Linear Models Spring 2017 • Question 4

Suppose \(X,Y\in L^1(\mathbb {P})\) satisfy \(\mathbb {E}[X\mid Y]=Y\) and \(\mathbb {E}[Y\mid X]=X\) almost surely. Prove that \(X=Y\) almost surely. (Hint: Prove first that \(\mathbb {E}[X-Y;X<q<Y]=0\) for all \(q\).)

Raw LaTeX

Suppose \(X,Y\in L^1(\Prob)\) satisfy \(\E[X\mid Y]=Y\) and
\(\E[Y\mid X]=X\) almost surely. Prove that \(X=Y\) almost surely. (Hint: Prove
first that \(\E[X-Y;X<q<Y]=0\) for all \(q\).)

Topic Tags

Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods