Math 6010 • Linear Models

Let \(X,X_1,X_2,\ldots\) be independent random variables, all
distributed according to the standard normal distribution.

\textbf{(a)} Prove that there exist finite constants \(c_1,c_2,x_0>0\), such that
\[
  \frac{c_1}{x}e^{-x^2/2}\leq \Prob\{X>x\}\leq
  \frac{c_2}{x}e^{-x^2/2}
  \qquad\text{for all }x\geq x_0.
\]

\textbf{(b)} Use the inequalities of part (a) in order to prove that if
\(X_1,X_2,\ldots\) are independent standard normal random variables then
\[
  \limsup_{n\to\infty}\frac{X_n}{\sqrt{2\ln n}}
  =
  -\liminf_{n\to\infty}\frac{X_n}{\sqrt{2\ln n}}
  =1 \qquad\text{a.s.}
\]