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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 10

Question

Math 6010 • Linear Models 2006 • Question 10

Let \(\{X_i\}_{i=1}^{\infty }\) be i.i.d. with the “standard Cauchy distribution.” That is, the density function of \(X_1\) is \[ f(x)=\frac {1}{\pi (1+x^2)} \qquad -\infty <x<\infty . \] Prove that \(\max _{1\le i\le n}X_i/n\) converges weakly. Identify the limit.

Raw LaTeX

Let \(\{X_i\}_{i=1}^{\infty}\) be i.i.d. with the ``standard Cauchy
distribution.'' That is, the density function of \(X_1\) is
\[
  f(x)=\frac{1}{\pi(1+x^2)}
  \qquad -\infty<x<\infty.
\]
Prove that \(\max_{1\le i\le n}X_i/n\) converges weakly. Identify the limit.

Topic Tags

Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits