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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 2

Question

Math 6010 • Linear Models Spring 2008 • Question 2

(a) State the central limit theorem as well as Kolmogorov’s 0-1 law.

(b) For arbitrary events \(A_n\), \(n\geq 1\), show that \(\inf _{n\geq 1}\mathbb {P}(A_n)>0\) implies \(\mathbb {P}(A_n\ \text {i.o.})>0\).

(c) Let \(X_1,X_2,\ldots \) be i.i.d. mean 0, variance 1 random variables, and put \(S_n=X_1+\cdots +X_n\) for each \(n\geq 1\). Show that \(\limsup _{n\to \infty }S_n=\infty \) a.s.

Raw LaTeX

(a) State the central limit theorem as well as Kolmogorov's 0-1 law.

(b) For arbitrary events \(A_n\), \(n\geq1\), show that
\(\inf_{n\geq1}\Prob(A_n)>0\) implies \(\Prob(A_n\ \text{i.o.})>0\).

(c) Let \(X_1,X_2,\ldots\) be i.i.d. mean 0, variance 1 random variables, and put
\(S_n=X_1+\cdots+X_n\) for each \(n\geq1\). Show that
\(\limsup_{n\to\infty}S_n=\infty\) a.s.

Topic Tags

Borel-Cantelli Lemmas Central Limit Theorems Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods