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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2011 • Question 3

Question

Math 6010 • Linear Models Spring 2011 • Question 3

Let \(X_1,\ldots ,X_n\) be independent identically distributed random variables with \(\mathbb {E} X_1=\mu \) and \(0<\operatorname {Var}(X_1)=\sigma ^2<\infty \). Let \[ Y_n=\sum _{1\leq i<j\leq n}X_iX_j. \] Find numerical sequences \(a_n\) and \(b_n\) such that \((Y_n-a_n)/b_n\) has a non-degenerate limit distribution.

Raw LaTeX

Let \(X_1,\ldots,X_n\) be independent identically distributed random
variables with \(\E X_1=\mu\) and \(0<\Var(X_1)=\sigma^2<\infty\). Let
\[
  Y_n=\sum_{1\leq i<j\leq n}X_iX_j.
\]
Find numerical sequences \(a_n\) and \(b_n\) such that \((Y_n-a_n)/b_n\) has a
non-degenerate limit distribution.

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Topic Tags

Covariance and Independence Estimator Distributions Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods