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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 3

Question

Math 6010 • Linear Models Fall 2008 • Question 3

Let \(\{X_n,n\geq 1\}\) be independent \(N(0,1)\) random variables. Show that \[ \Prob \left (\limsup \frac {|X_n|}{\sqrt {\log n}}=\sqrt {2}\right )=1. \] Hint: \(1-\Phi (x)\) is asymptotic to \(\phi (x)/x\) as \(x\to \infty \).

Raw LaTeX

Let \(\{X_n,n\geq1\}\) be independent \(\N(0,1)\) random variables.
Show that
\[
  \Prob\left(\limsup \frac{|X_n|}{\sqrt{\log n}}=\sqrt{2}\right)=1.
\]
Hint: \(1-\Phi(x)\) is asymptotic to \(\phi(x)/x\) as \(x\to\infty\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods