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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 \(\limsup _{n\to \infty }S_n/\sqrt n=\infty \) 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
\(\sigma^2\in(0,\infty)\).

(a) Use the central limit theorem as well as Kolmogorov's 0-1 law to conclude
that \(\limsup_{n\to\infty}S_n/\sqrt n=\infty\) a.s.

(b) Show that \(S_n/\sqrt n\) does not converge in probability.

Topic Tags

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