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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 5

Question

Math 6010 • Linear Models 1994 • Question 5

Let \(X_1,X_2,\cdots \) be independent, identically distributed normal random variables. Find two numerical sequences, \(a_n\) and \(b_n\), such that \[ \frac {\max _{1\le i\le n}X_i-a_n}{b_n} \] converges in distribution to a non-degenerate random variable.

Raw LaTeX

Let \(X_1,X_2,\cdots\) be independent, identically distributed normal
random variables. Find two numerical sequences, \(a_n\) and \(b_n\), such that
\[
\frac{\max_{1\le i\le n}X_i-a_n}{b_n}
\]
converges in distribution to a non-degenerate random variable.

Topic Tags

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