Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2018 • Question 10

Question

Math 6010 • Linear Models Spring 2018 • Question 10

Give an example of a sequence of random variables \(X_n\) with densities \(f_n(x)\) so that \(X_n\) converges weakly as \(n\to \infty \) to a uniform random variable on \([0,1]\), but \(f_n(x)\) does not converge to 1 for any \(x\in [0,1]\).

Raw LaTeX

Give an example of a sequence of random variables \(X_n\) with
densities \(f_n(x)\) so that \(X_n\) converges weakly as \(n\to\infty\) to a uniform
random variable on \([0,1]\), but \(f_n(x)\) does not converge to 1 for any
\(x\in[0,1]\).

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits