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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2017 • Question 8

Question

Math 6010 • Linear Models Spring 2017 • Question 8

Suppose that \(X_1,X_2,\ldots \) are i.i.d. exponential random variables with parameter \(\lambda >0\). Prove that \[ \overline {\lim }_{n\to \infty }\left (\frac {X_n}{\log n}\right )=\lambda ^{-1} \qquad \text {and}\qquad \underline {\lim }_{n\to \infty }\left (\frac {\log X_n}{\log n}\right )=-1. \]

Raw LaTeX

Suppose that \(X_1,X_2,\ldots\) are i.i.d. exponential random
variables with parameter \(\lambda>0\). Prove that
\[
  \overline{\lim}_{n\to\infty}\left(\frac{X_n}{\log n}\right)=\lambda^{-1}
  \qquad\text{and}\qquad
  \underline{\lim}_{n\to\infty}\left(\frac{\log X_n}{\log n}\right)=-1.
\]

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Topic Tags

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