Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 5

Question

Math 6010 • Linear Models 2006 • Question 5

Let \(\{X_n\}_{n=1}^{\infty }\) be a sequence of independent (but not identically distributed) random variables, such that for all \(n\ge 1\), \[ \mathbb {P}\{X_n=1\}=\mathbb {P}\{X_n=-1\}=\frac {1}{2}-\frac {1}{2n^2}, \qquad \mathbb {P}\{X_n=n\}=\mathbb {P}\{X_n=-n\}=\frac {1}{2n^2}. \] Prove that \(n^{-1/2}\sum _{i=1}^{n}X_i\Rightarrow N(0,\sigma ^2)\), and compute \(\sigma ^2\) explicitly.

Raw LaTeX

Let \(\{X_n\}_{n=1}^{\infty}\) be a sequence of independent (but not
identically distributed) random variables, such that for all \(n\ge 1\),
\[
  \Prob\{X_n=1\}=\Prob\{X_n=-1\}=\frac{1}{2}-\frac{1}{2n^2}, \qquad
  \Prob\{X_n=n\}=\Prob\{X_n=-n\}=\frac{1}{2n^2}.
\]
Prove that \(n^{-1/2}\sum_{i=1}^{n}X_i\Rightarrow N(0,\sigma^2)\), and compute
\(\sigma^2\) explicitly.

Topic Tags

Covariance and Independence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods