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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 1

Question

Math 6010 • Linear Models Spring 2015 • Question 1

\(X_1,X_2,\ldots ,X_n\) be independent identically distributed positive random variables. Show that \[ \frac {1}{n}\max _{1\leq k\leq n} X_k \to 0 \qquad \text {a.s.} \] if and only if \(\mathbb {E} X_1<\infty \).

Raw LaTeX

\(X_1,X_2,\ldots,X_n\) be independent identically distributed
positive random variables. Show that
\[
  \frac{1}{n}\max_{1\leq k\leq n} X_k \to 0
  \qquad\text{a.s.}
\]
if and only if \(\E X_1<\infty\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Laws of Large Numbers Uniform Integrability and Truncation Methods