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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 6

Question

Math 6010 • Linear Models Fall 2016 • Question 6

Suppose \(\{X_n\}_{n=1}^{\infty }\) are i.i.d. exponential random variables with mean one; that is, \(\mathbb {P}\{X_n>x\}=e^{-x}\) for all \(x>0\). Prove that \[ \mathbb {P}\left \{\limsup _{n\to \infty }\frac {X_n}{\log n}=1\right \}=1. \]

Raw LaTeX

Suppose \(\{X_n\}_{n=1}^{\infty}\) are i.i.d. exponential random
variables with mean one; that is, \(\Prob\{X_n>x\}=e^{-x}\) for all \(x>0\).
Prove that
\[
  \Prob\left\{\limsup_{n\to\infty}\frac{X_n}{\log n}=1\right\}=1.
\]

Topic Tags

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