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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 8

Question

Math 6010 • Linear Models 2005 • Question 8

Prove that whenever \(X_n\Rightarrow X\) and \(Y_n\to c\) in probability for a non-random \(c>0\), then \(X_n/Y_n\Rightarrow c^{-1}X\). Use this to prove the following: If \(\xi _1,\xi _2,\ldots \) are i.i.d. with \(\mathbb {E}[\xi _1]=\mu \) and \(\operatorname {Var}(\xi _1)=\sigma ^2\) then \[ \frac {S_n-n\mu }{\sqrt {\sum _{i=1}^{n}(\xi _i-\bar {\xi }_n)^2}} \Rightarrow N(0,1). \] where \(S_n=\xi _1+\cdots +\xi _n\), and \(\bar {\xi }_n=S_n/n\).

Raw LaTeX

Prove that whenever \(X_n\Rightarrow X\) and
\(Y_n\to c\) in probability for a non-random \(c>0\), then
\(X_n/Y_n\Rightarrow c^{-1}X\). Use this to prove the following: If
\(\xi_1,\xi_2,\ldots\) are i.i.d. with \(\E[\xi_1]=\mu\) and
\(\Var(\xi_1)=\sigma^2\) then
\[
  \frac{S_n-n\mu}{\sqrt{\sum_{i=1}^{n}(\xi_i-\bar{\xi}_n)^2}}
  \Rightarrow N(0,1).
\]
where \(S_n=\xi_1+\cdots+\xi_n\), and \(\bar{\xi}_n=S_n/n\).

Topic Tags

Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods