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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 2

Question

Math 6010 • Linear Models 2000 • Question 2

Let \(X\) be such that \(\mathbb {E}[X^+]=\infty \) and \(\mathbb {E}[X^-]<\infty \), and let \(X_1,X_2,X_3,\ldots \) be i.i.d. with common distribution that of \(X\). With \(S_n=X_1+X_2+\cdots +X_n\), show that \(\lim _{n\to \infty }S_n/n=\infty \) a.s.

Raw LaTeX

Let \(X\) be such that \(\E[X^+]=\infty\) and \(\E[X^-]<\infty\), and let
\(X_1,X_2,X_3,\ldots\) be i.i.d. with common distribution that of \(X\). With
\(S_n=X_1+X_2+\cdots+X_n\), show that
\(\lim_{n\to\infty}S_n/n=\infty\) a.s.

Topic Tags

Convergence in Probability and Almost Sure Convergence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods