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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 7

Question

Math 6010 • Linear Models 2005 • Question 7

Suppose \(X_1,X_2,\ldots \) have common mean \(\mu \) and variances \(\sigma _1^2,\sigma _2^2,\ldots \). Prove that if \(\sup _n\sigma _n^2<\infty \) and \(\lim _{|i-j|\to \infty }\mathbb {E}[X_iX_j]=0\), then \((X_1+\cdots +X_n)/n\) converges to \(\mu \) in probability.

Raw LaTeX

Suppose \(X_1,X_2,\ldots\) have common mean \(\mu\) and variances
\(\sigma_1^2,\sigma_2^2,\ldots\). Prove that if
\(\sup_n\sigma_n^2<\infty\) and
\(\lim_{|i-j|\to\infty}\E[X_iX_j]=0\), then
\((X_1+\cdots+X_n)/n\) converges to \(\mu\) in probability.

Topic Tags

Analysis of Variance Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods