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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 2

Question

Math 6010 • Linear Models Fall 2018 • Question 2

Let \(\{X_i:i\geq 1\}\) be a sequence of non-negative, i.i.d. random variables such \(\mathbb {E}[X_1]=\infty \). Prove that \[ \lim _{n\to \infty }\frac {X_1+\cdots +X_n}{n}=\infty \qquad \text {almost surely.} \]

Raw LaTeX

Let \(\{X_i:i\geq1\}\) be a sequence of non-negative, i.i.d. random
variables such \(\E[X_1]=\infty\). Prove that
\[
  \lim_{n\to\infty}\frac{X_1+\cdots+X_n}{n}=\infty
  \qquad\text{almost surely.}
\]

Topic Tags

Convergence in Probability and Almost Sure Convergence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods