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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2011 • Question 4

Question

Math 6010 • Linear Models Spring 2011 • Question 4

Let \(f\) be a continuous and bounded function on \([0,\infty )\). Compute \[ \lim _{n\to \infty }\int _0^\infty \cdots \int _0^\infty f\!\left (\frac {x_1+\cdots +x_n}{n}\right ) e^{-(x_1+\cdots +x_n)}\,dx_1\cdots x_n. \]

Raw LaTeX

Let \(f\) be a continuous and bounded function on \([0,\infty)\).
Compute
\[
  \lim_{n\to\infty}\int_0^\infty\cdots\int_0^\infty
  f\!\left(\frac{x_1+\cdots+x_n}{n}\right)
  e^{-(x_1+\cdots+x_n)}\,dx_1\cdots x_n.
\]

Topic Tags

Uniform Integrability and Truncation Methods