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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 8

Question

Math 6010 • Linear Models Fall 2008 • Question 8

Let \(X_1,X_2,\ldots \) and \(Y_1,Y_2,\ldots \) denote random variables on a common probability space. Suppose \(X_n\) converges to 0 in probability as \(n\to \infty \), and \(Y_n\) converges to \(Y\) weakly. Prove that \((X_n,Y_n)\) converges weakly to \((0,Y)\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) and \(Y_1,Y_2,\ldots\) denote random variables on
a common probability space. Suppose \(X_n\) converges to 0 in probability as
\(n\to\infty\), and \(Y_n\) converges to \(Y\) weakly. Prove that \((X_n,Y_n)\)
converges weakly to \((0,Y)\).

Topic Tags

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