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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 9

Question

Math 6010 • Linear Models Fall 2018 • Question 9

Let \(Z_1,Z_2,\ldots \) be iid \(N(0,1)\) random variables. First prove Mills’ ratio \[ \mathbb {P}(Z_i>\lambda ) \leq \frac {1}{\lambda \sqrt {2\pi }}e^{-\lambda ^2/2} \] for any \(\lambda >0\). Use this to show that for any \(\epsilon >0\) \[ \mathbb {P}\left ( \limsup _{n\to \infty } \frac {\max _{k\leq n}Z_k}{\sqrt {(2-\epsilon )\log n}} >1 \right )=1. \]

Raw LaTeX

Let \(Z_1,Z_2,\ldots\) be iid \(N(0,1)\) random variables. First prove
Mills' ratio
\[
  \Prob(Z_i>\lambda)
  \leq
  \frac{1}{\lambda\sqrt{2\pi}}e^{-\lambda^2/2}
\]
for any \(\lambda>0\). Use this to show that for any \(\epsilon>0\)
\[
  \Prob\left(
    \limsup_{n\to\infty}
    \frac{\max_{k\leq n}Z_k}{\sqrt{(2-\epsilon)\log n}}
    >1
  \right)=1.
\]

Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods