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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2010 • Question 2

Question

Math 6010 • Linear Models Fall 2010 • Question 2

Let \(X_1,X_2,\ldots \) be independent identically distributed standard normal random variables. Show that \[ \limsup _{n\to \infty }\frac {1}{\sqrt {\log n}}|X_n|=c \quad \text {almost surely} \] and compute the value of \(c\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent identically distributed standard
normal random variables. Show that
\[
  \limsup_{n\to\infty}\frac{1}{\sqrt{\log n}}|X_n|=c
  \quad\text{almost surely}
\]
and compute the value of \(c\).

Topic Tags

Central Limit Theorems Convergence in Probability and Almost Sure Convergence Covariance and Independence Uniform Integrability and Truncation Methods