Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 9

Question

Math 6010 • Linear Models Fall 2013 • Question 9

Let \(X\) be an integer-valued random variable that is non-degenerate; this means that there exist at least two integers \(n,m\) such that \(\mathbb {P}\{X=n\}\) and \(\mathbb {P}\{X=m\}\) are both strictly positive. Prove that the set \[ \mathcal {V}:=\{t\in \mathbf {R}:|\mathbb {E}[e^{itX}]|=1\} \] has zero Lebesgue measure. [Hint: If \(t\in \mathcal {V}\) then \(Z:=e^{it(X-Y)}\) has mean one, where \(Y\) is an independent copy of \(X\). Proceed by proving that \(\mathbb {E}(|Z-1|^2)=0\).]

Raw LaTeX

Let \(X\) be an integer-valued random variable that is
non-degenerate; this means that there exist at least two integers \(n,m\) such
that \(\Prob\{X=n\}\) and \(\Prob\{X=m\}\) are both strictly positive. Prove that
the set
\[
  \mathcal{V}:=\{t\in\mathbf{R}:|\E[e^{itX}]|=1\}
\]
has zero Lebesgue measure. [Hint: If \(t\in\mathcal{V}\) then
\(Z:=e^{it(X-Y)}\) has mean one, where \(Y\) is an independent copy of \(X\). Proceed
by proving that \(\E(|Z-1|^2)=0\).]

Topic Tags

Covariance and Independence Lebesgue Differentiation via Martingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods