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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2019 • Question 10

Question

Math 6010 • Linear Models Spring 2019 • Question 10

Let \(g:\mathbb {R}\to \mathbb {R}\) be bounded and continuous and fix \(\lambda >0\). Use properties of the Poisson distribution to show that \[ \sum _{k=0}^{\infty }g(k/n)\frac {(n\lambda )^k}{k!}e^{-n\lambda } \to g(\lambda ) \qquad \text {as } n\to \infty . \]

Raw LaTeX

Let \(g:\mathbb{R}\to\mathbb{R}\) be bounded and continuous and fix
\(\lambda>0\). Use properties of the Poisson distribution to show that
\[
  \sum_{k=0}^{\infty}g(k/n)\frac{(n\lambda)^k}{k!}e^{-n\lambda}
  \to g(\lambda)
  \qquad\text{as } n\to\infty.
\]

Topic Tags

Estimator Distributions Multivariate Normal Distributions Quadratic Forms and Chi-Squared Distributions Uniform Integrability and Truncation Methods