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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 1

Question

Math 6010 • Linear Models 1996 • Question 1

Prove Cantelli’s theorem: If \(X_1,X_2,\ldots \) are independent (but not necessarily identically distributed), mean zero, random variables with \(\sup _n \mathbb {E}[X_n^4]<\infty \), then \(S_n/n\to 0\) a.s. as \(n\to \infty \).

Raw LaTeX

Prove Cantelli's theorem: If \(X_1,X_2,\ldots\) are independent (but
not necessarily identically distributed), mean zero, random variables with
\(\sup_n \E[X_n^4]<\infty\), then \(S_n/n\to 0\) a.s. as \(n\to\infty\).

Topic Tags

Borel-Cantelli Lemmas Convergence in Probability and Almost Sure Convergence Covariance and Independence Uniform Integrability and Truncation Methods