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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 6

Question

Math 6010 • Linear Models Fall 2009 • Question 6

The Cauchy distribution has density \(f(x):=1/[\pi (1+x^2)]\) for \(-\infty <x<\infty \) and characteristic function \(\varphi (t):=e^{-|t|}\) for \(-\infty <t<\infty \). Let \(X_1,X_2,\ldots \) be i.i.d. Cauchy, put \(S_n:=X_1+\cdots +X_n\) for each \(n\geq 1\). Show that \(S_n/n\) does not converge almost surely or in probability, but does converge in distribution.

Raw LaTeX

The Cauchy distribution has density
\(f(x):=1/[\pi(1+x^2)]\) for \(-\infty<x<\infty\) and characteristic function
\(\varphi(t):=e^{-|t|}\) for \(-\infty<t<\infty\). Let \(X_1,X_2,\ldots\) be i.i.d.
Cauchy, put \(S_n:=X_1+\cdots+X_n\) for each \(n\geq1\). Show that \(S_n/n\) does
not converge almost surely or in probability, but does converge in distribution.

Topic Tags

Characteristic Functions and Fourier Inversion Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Probability Inequalities and Tail Bounds