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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2019 • Question 5

Question

Math 6010 • Linear Models Spring 2019 • Question 5

Let \(X_n\) be a sequence of independent random variables with \(\mathbb {P}(X_n=1)=1/n=1-\mathbb {P}(X_n=0)\). Prove that \[ \left (\sum _{i=1}^{n}X_i-\log n\right )/\sqrt {\log n} \] converges weakly to a standard normal.

Raw LaTeX

Let \(X_n\) be a sequence of independent random variables with
\(\Prob(X_n=1)=1/n=1-\Prob(X_n=0)\). Prove that
\[
  \left(\sum_{i=1}^{n}X_i-\log n\right)/\sqrt{\log n}
\]
converges weakly to a standard normal.

Topic Tags

Central Limit Theorems Covariance and Independence Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits