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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 2

Question

Math 6010 • Linear Models Fall 2017 • Question 2

Let \(X_n\) be i.i.d. random variables with mean \(\mu \) and variance \(\sigma ^2<\infty \). Prove (without using the law of the iterated logarithm) that we have almost surely \[ \overline {\lim }_{n\to \infty }\frac {S_n-n\mu }{\sigma \sqrt {n}}=\infty \qquad \text {and}\qquad \underline {\lim }_{n\to \infty }\frac {S_n-n\mu }{\sigma \sqrt {n}}=-\infty . \]

Raw LaTeX

Let \(X_n\) be i.i.d. random variables with mean \(\mu\) and variance
\(\sigma^2<\infty\). Prove (without using the law of the iterated logarithm) that
we have almost surely
\[
  \overline{\lim}_{n\to\infty}\frac{S_n-n\mu}{\sigma\sqrt{n}}=\infty
  \qquad\text{and}\qquad
  \underline{\lim}_{n\to\infty}\frac{S_n-n\mu}{\sigma\sqrt{n}}=-\infty.
\]

Topic Tags

Central Limit Theorems Convergence in Probability and Almost Sure Convergence Law of the Iterated Logarithm Uniform Integrability and Truncation Methods