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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 6

Question

Math 6010 • Linear Models Spring 2009 • Question 6

Suppose \(X_n\Rightarrow X\) as \(n\to \infty \), and \(\sup _{n\geq 1}\mathbb {E}(X_n^2)<\infty \). Prove that \(X\in L^1(\mathbb {P})\) and \(\lim _{n\to \infty }\mathbb {E} X_n=\mathbb {E} X\).

Raw LaTeX

Suppose \(X_n\Rightarrow X\) as \(n\to\infty\), and
\(\sup_{n\geq1}\E(X_n^2)<\infty\). Prove that \(X\in L^1(\Prob)\) and
\(\lim_{n\to\infty}\E X_n=\E X\).

Topic Tags

Uniform Integrability and Truncation Methods