Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 3

Question

Math 6010 • Linear Models 1993 • Question 3
Let \(X_1,X_2,\ldots\) be i.i.d. mean 0, variance \(\sigma^2\in(0,\infty)\). **(a)** Use the central limit theorem as well as Kolmogorov's 0--1 law to conclude that \(\limsup S_n/\sqrt{n}=\infty\) a.s. **(b)** Show that \(S_n/\sqrt{n}\) does not converge in probability.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. mean 0, variance \(\sigma^2\in(0,\infty)\).
\begin{parts}
\item Use the central limit theorem as well as Kolmogorov's 0--1 law to conclude that \(\limsup S_n/\sqrt{n}=\infty\) a.s.
\item Show that \(S_n/\sqrt{n}\) does not converge in probability.
\end{parts}

Topic Tags

Analysis of Variance Central Limit Theorems Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods