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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 3

Question

Math 6010 • Linear Models Spring 2009 • Question 3

For \(n=1,2,\ldots \), let \(X_n\) have density \[ f_n(x):=\frac {n}{\pi (1+n^2x^2)},\qquad -\infty <x<\infty . \] (a) Does \(X_n\) converge to 0 in probability?

(b) Does \(X_n\) converge to 0 in \(L^1(\mathbb {P})\)?

(c) If \(X_1,X_2,\ldots \) are independent, does \(X_n\) converge to 0 a.s.?

Raw LaTeX

For \(n=1,2,\ldots\), let \(X_n\) have density
\[
  f_n(x):=\frac{n}{\pi(1+n^2x^2)},\qquad -\infty<x<\infty.
\]
\textbf{(a)} Does \(X_n\) converge to 0 in probability?

\textbf{(b)} Does \(X_n\) converge to 0 in \(L^1(\Prob)\)?

\textbf{(c)} If \(X_1,X_2,\ldots\) are independent, does \(X_n\) converge to 0 a.s.?

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods