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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 5

Question

Math 6010 • Linear Models 2005 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with distribution function \[ F(t)=\frac {1}{1+e^{-t}},\qquad -\infty <t<\infty . \] Show that \[ Y_n=X_{n,n}-\log n \] converges in distribution, where \(X_{n,n}=\max \{X_1,X_2,\ldots ,X_n\}\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed random variables with distribution function
\[
F(t)=\frac{1}{1+e^{-t}},\qquad -\infty<t<\infty.
\]
Show that
\[
Y_n=X_{n,n}-\log n
\]
converges in distribution, where \(X_{n,n}=\max\{X_1,X_2,\ldots,X_n\}\).

Topic Tags

Covariance and Independence Estimator Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits