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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 6

Question

Math 6010 • Linear Models 2004 • Question 6

Suppose \(\{X_n\}_{n=1}^{\infty }\) and \(\{Y_n\}_{n=1}^{\infty }\) are random variables on the same probability space \((\Omega ,\mathcal {F},\mathbb {P})\). Suppose also that \(X_n\Rightarrow X\) and \(Y_n\Rightarrow Y\). True or false: \((X_n,Y_n)\Rightarrow (X,Y)\). [If true, then prove it. If false, then construct a counter-example.]

Raw LaTeX

Suppose \(\{X_n\}_{n=1}^{\infty}\) and \(\{Y_n\}_{n=1}^{\infty}\) are random
variables on the same probability space \((\Omega,\mathcal{F},\Prob)\). Suppose also
that \(X_n\Rightarrow X\) and \(Y_n\Rightarrow Y\). True or false:
\((X_n,Y_n)\Rightarrow(X,Y)\). [If true, then prove it. If false, then construct a
counter-example.]

Topic Tags

Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods