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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2019 • Question 1

Question

Math 6010 • Linear Models Fall 2019 • Question 1

Assume that \(A_i\), \(i=1,2,\ldots \), are independent events. Prove that \[ \frac {1}{n}\sum _{i=1}^{n}\left (\mathbf {1}_{A_i}-\mathbb {P}(A_i)\right ) \] converges to zero in probability as \(n\to \infty \), where where \(\mathbf {1}_{A_i}\) is the indicator function of the event \(A_i\).

Raw LaTeX

Assume that \(A_i\), \(i=1,2,\ldots\), are independent events. Prove
that
\[
  \frac{1}{n}\sum_{i=1}^{n}\left(\mathbf{1}_{A_i}-\Prob(A_i)\right)
\]
converges to zero in probability as \(n\to\infty\), where where
\(\mathbf{1}_{A_i}\) is the indicator function of the event \(A_i\).

Topic Tags

Classical Probability and Inclusion-Exclusion Covariance and Independence Laws of Large Numbers Uniform Integrability and Truncation Methods