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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 3

Question

Math 6010 • Linear Models 1996 • Question 3

Let \((\Omega ,\mathcal {F},P)\) be a probability space, let \(\mathcal {G}\) be a sub-\(\sigma \)-field of \(\mathcal {F}\), and let \(M\) be the closed subspace of \(L^2(\Omega ,\mathcal {F},P)\) consisting of the \(\mathcal {G}\)-measurable functions in \(L^2\). Show that \(T(X)=E[X\mid \mathcal {G}]\) coincides with the orthogonal projection of \(L^2(\Omega ,\mathcal {F},P)\) onto \(M\).

Raw LaTeX

Let \((\Omega,\mathcal{F},P)\) be a probability space, let \(\mathcal{G}\) be a sub-\(\sigma\)-field of \(\mathcal{F}\), and let \(M\) be the closed subspace of \(L^2(\Omega,\mathcal{F},P)\) consisting of the \(\mathcal{G}\)-measurable functions in \(L^2\). Show that \(T(X)=E[X\mid\mathcal{G}]\) coincides with the orthogonal projection of \(L^2(\Omega,\mathcal{F},P)\) onto \(M\).

Topic Tags

L2 Convergence and Covariance Bounds Measure-Theoretic Probability Orthogonal Projections Uniform Integrability and Truncation Methods