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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 6

Question

Math 6010 • Linear Models Spring 2015 • Question 6

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential(1) random variables. Let \(k\geq 1\). Show that \[ \mathbb {P}\{X_n>\phi _{\varepsilon }(n)\ \text {i.o.}\} = \begin {cases} 0, & \text {if }\varepsilon >0,\\ 1, & \text {if }\varepsilon \leq 0, \end {cases} \] where \[ \phi _{\varepsilon }(n)=\log n+\log _{(2)}(n)+\cdots + (1+\varepsilon )\log _{(k)}(n), \] and \(\log _{(1)}(n)=\log n\), \(\log _{(k)}(n)=\log (\log _{(k-1)}(n))\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
exponential(1) random variables. Let \(k\geq1\). Show that
\[
  \Prob\{X_n>\phi_{\varepsilon}(n)\ \text{i.o.}\} =
  \begin{cases}
    0, & \text{if }\varepsilon>0,\\
    1, & \text{if }\varepsilon\leq0,
  \end{cases}
\]
where
\[
  \phi_{\varepsilon}(n)=\log n+\log_{(2)}(n)+\cdots+
  (1+\varepsilon)\log_{(k)}(n),
\]
and \(\log_{(1)}(n)=\log n\),
\(\log_{(k)}(n)=\log(\log_{(k-1)}(n))\).

Topic Tags

Borel-Cantelli Lemmas Covariance and Independence Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods