Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 1

Question

Math 6010 • Linear Models Spring 2020 • Question 1

Let \(X_1,X_2,\ldots \), and \(X\) be real-valued random variables defined on a common probability space. Note they do NOT have to be independent or identically distributed. Prove that \[ \sum _{n=1}^{\infty }\mathbb {E}[|X_n-X|]<\infty \Longrightarrow X_n\to X \quad \text {almost surely.} \]

Raw LaTeX

Let \(X_1,X_2,\ldots\), and \(X\) be real-valued random variables
defined on a common probability space. Note they do NOT have to be independent
or identically distributed. Prove that
\[
  \sum_{n=1}^{\infty}\E[|X_n-X|]<\infty
  \Longrightarrow
  X_n\to X
  \quad\text{almost surely.}
\]

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods