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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

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with probability mass function \[ P\{X_i=1\}=\theta \quad \text {and}\quad P\{X_i=0\}=1-\theta . \] Let: \[ \bar {X}_n=\frac {1}{n}\sum _{1\leq i\leq n} X_i \] and \(h(t)=t^3-t\). Find the sequences \(a_n\) and \(b_n\) such that \(a_n\bigl (h(\bar {X}_n)-h(\theta )-b_n\bigr )\) has a non-degenerate limit distribution.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed random variables with probability mass function
\[
P\{X_i=1\}=\theta\quad\text{and}\quad P\{X_i=0\}=1-\theta.
\]
Let:
\[
\bar{X}_n=\frac{1}{n}\sum_{1\leq i\leq n} X_i
\]
and \(h(t)=t^3-t\). Find the sequences \(a_n\) and \(b_n\) such that \(a_n\bigl(h(\bar{X}_n)-h(\theta)-b_n\bigr)\) has a non-degenerate limit distribution.

Topic Tags

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