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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 5

Question

Math 6010 • Linear Models Spring 2016 • Question 5

Suppose \(g\) and \(h\) are continuous on \(\mathbf {R}\) with \(g>0\) and \(|h(x)|/g(x)\to 0\) as \(|x|\to \infty \). If \(X_n\Rightarrow X\) and \(\sup _n\mathbb {E}[g(X_n)]<\infty \), show that \(\mathbb {E}[h(X_n)]\to \mathbb {E}[h(X)]\).

Raw LaTeX

Suppose \(g\) and \(h\) are continuous on \(\mathbf{R}\) with \(g>0\) and
\(|h(x)|/g(x)\to0\) as \(|x|\to\infty\). If \(X_n\Rightarrow X\) and
\(\sup_n\E[g(X_n)]<\infty\), show that \(\E[h(X_n)]\to\E[h(X)]\).

Topic Tags

Uniform Integrability and Truncation Methods