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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 8

Question

Math 6010 • Linear Models Spring 2010 • Question 8

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \) and finite variance. Show that \[ U_n:=\binom {n}{2}^{-1}\sum _{1\leq i<j\leq n}X_iX_j \] converges in probability to \(\mu ^2\) as \(n\to \infty \).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. with mean \(\mu\) and finite variance.
Show that
\[
  U_n:=\binom{n}{2}^{-1}\sum_{1\leq i<j\leq n}X_iX_j
\]
converges in probability to \(\mu^2\) as \(n\to\infty\).

Topic Tags

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