Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2014 • Question 5

Question

Math 6010 • Linear Models Fall 2014 • Question 5

Let \(X_1,X_2,\ldots \) be independent, identically distributed random variables with the following common distribution: \[ \mathbb {P}\{X_1=k\}=\frac {3}{\pi ^2k^2} \qquad \text {for } k=\pm 1,\pm 2,\ldots . \]

(a) Prove that \(\mathbb {E}\exp (itX_1)=1-(3|t|/\pi )+o(|t|)\) as \(t\to 0\). You may use, without proof, the fact that \[ \int _0^{\infty }(1-\cos \theta )\theta ^{-2}\,d\theta =\pi /2. \]

(b) Use the preceding in order to prove that \(n^{-1}(X_1+\cdots +X_n)\) converges weakly to a “Cauchy random variable \(Y\) with scale parameter \(3/\pi \).” That is, the characteristic function of \(Y\) is \[ \mathbb {E}\exp (itY)=\exp (-3|t|/\pi ) \qquad \text {for all }t\in \mathbb {R}. \] You may use, without proof, the fact that the latter is a characteristic function. This matter is the topic of Problem 8 below.

Let \(X_1,X_2,\ldots\) be independent, identically distributed
random variables with the following common distribution:
\[
  \Prob\{X_1=k\}=\frac{3}{\pi^2k^2}
  \qquad\text{for } k=\pm1,\pm2,\ldots .
\]

\textbf{(a)} Prove that
\(\E\exp(itX_1)=1-(3|t|/\pi)+o(|t|)\) as \(t\to0\).
You may use, without proof, the fact that
\[
  \int_0^{\infty}(1-\cos\theta)\theta^{-2}\,d\theta=\pi/2.
\]

\textbf{(b)} Use the preceding in order to prove that
\(n^{-1}(X_1+\cdots+X_n)\) converges weakly to a ``Cauchy random variable \(Y\)
with scale parameter \(3/\pi\).'' That is, the characteristic function of \(Y\) is
\[
  \E\exp(itY)=\exp(-3|t|/\pi)
  \qquad\text{for all }t\in\R.
\]
You may use, without proof, the fact that the latter is a characteristic
function. This matter is the topic of Problem 8 below.

Topic Tags

Characteristic Functions and Fourier Inversion Covariance and Independence Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits