Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 9

Question

Math 6010 • Linear Models Spring 2008 • Question 9

(5 points) Let \(\zeta _n\) and \(\eta _n\) be two sequences of random variables. Prove or give counterexamples to the following statements:

9a) If \(\zeta _n\xrightarrow {(d)}\zeta \) and \(\eta _n\xrightarrow {(d)}\eta \), then \(\zeta _n+\eta _n\xrightarrow {(d)}\zeta +\eta \).

9b) If \(\zeta _n\xrightarrow {P}\zeta \) and \(\eta _n\xrightarrow {P}\eta \), then \(\zeta _n+\eta _n\xrightarrow {P}\zeta +\eta \).

9c) If \(\zeta _n\xrightarrow {(d)}\zeta \) and \(\eta _n\xrightarrow {(d)}\eta \), then \(\zeta _n+\eta _n\xrightarrow {P}\zeta +\eta \).

(\(\xrightarrow {(d)}\) and \(\xrightarrow {P}\) denote convergence in distribution and in probability, respectively.)

(5 points) Let \(\zeta_n\) and \(\eta_n\) be two sequences of random
variables. Prove or give counterexamples to the following statements:

\textbf{9a)} If \(\zeta_n\dto\zeta\) and
\(\eta_n\dto\eta\), then
\(\zeta_n+\eta_n\dto\zeta+\eta\).

\textbf{9b)} If \(\zeta_n\pto\zeta\) and
\(\eta_n\pto\eta\), then
\(\zeta_n+\eta_n\pto\zeta+\eta\).

\textbf{9c)} If \(\zeta_n\dto\zeta\) and
\(\eta_n\dto\eta\), then
\(\zeta_n+\eta_n\pto\zeta+\eta\).

(\(\dto\) and \(\pto\) denote convergence in distribution and in probability,
respectively.)

Topic Tags

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